# Null projections and noncommutative function theory in operator algebras

David P. Blecher  Department of Mathematics, University of Houston
<EMAIL_ADDRESS>and  Raphaël Clouâtre  Department of Mathematics,
University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2
<EMAIL_ADDRESS>
###### Abstract.

We study projections in the bidual of a $\mathrm{C}^{*}$-algebra $B$ that are
null with respect to a subalgebra $A$, that is projections $p\in B^{**}$
satisfying $|\varphi|(p)=0$ for every $\varphi\in B^{*}$ annihilating $A$. In
the separable case, $A$-null projections are precisely the peak projections in
the bidual of $A$ at which the subalgebra $A$ interpolates the entire
$\mathrm{C}^{*}$-algebra $B$. These are analogues of null sets in classical
function theory, on which several profound results rely. This motivates the
development of a noncommutative variant, which we use to find appropriate
‘quantized’ versions of some of these classical facts. Through a delicate
generalization of a theorem of Varopoulos, we show that, roughly speaking,
sufficiently regular interpolation projections are null precisely when their
atomic parts are. As an application, we give alternative proofs and
sharpenings of some recent peak-interpolation results of Davidson and Hartz
for algebras on Hilbert function spaces, also illuminating thereby how earlier
noncommutative peak-interpolation theory may be applied. In another direction,
given a convex subset of the state space of $B$, we characterize when the
associated Riesz projection is null. This is then applied to various important
topics in noncommutative function theory, such as the F.& M. Riesz property,
the existence of Lebesgue decompositions, the description of Henkin
functionals, and Arveson’s noncommutative Hardy spaces (maximal subdiagonal
algebras).

R.C. was partially supported by an NSERC Discovery Grant. D.B. was partially
supported by a Simons Foundation Travel/Collaboration grant and NSF Grant
DMS-2154903.

##  1\. Introduction

Let $B$ be a unital $\mathrm{C}^{*}$-algebra. A projection $u$ in the von
Neumann algebra $B^{**}$ is _open_ if it is the increasing weak-$*$ limit of a
net of positive contractions in $B$. Likewise, a projection $q\in B^{**}$ is
closed if $1-q$ is open. These notions of non-commutative topology were
introduced and developed by Akemann [akemann1969, akemann1970left].

Next, let $A\subset B$ be a unital norm-closed subalgebra. A closed projection
$q\in B^{**}$ is said to be an _$A$ -peak_ projection if there is a
contraction $a\in A$ such that $aq=q$ and $\|ap\|<1$ for every closed
projection $p\in B^{**}$ orthogonal to $q$. This is readily seen to be a
generalization of the classical idea of a peak set for a function algebra
[gamelin1969]. The foundations of the theory of non-commutative peaking in
this sense have been laid out in a series of papers by the first author and
collaborators (see e.g. [hay2007, BHN2008, BR2011, BN2012, BR2013,
blecher2013], the last of these being in part a survey). We shall refer to
this theory below as noncommutative peak-interpolation. An important result
proved in these papers is that, when $B$ is separable, a closed projection $q$
is $A$-peak precisely when $q$ lies in $A^{\perp\perp}$. Outside of the
separable setting, closed projections in $A^{\perp\perp}$ are the infima of
$A$-peak projections. These are called the generalized $A$-peak projections
(or $p$-projections) and they have properties similar to the peak projections.

Peak sets are a key tool in classical function theory. For example, when
$B=\mathrm{C}(X)$ is separable, it is known that the collection of $A$-peak
points (i.e. the minimal $A$-peak projections in $B^{**}$) coincides with the
minimal boundary of $A$ [bishop1959, bishop1959bdry]. By analogy, in the
general setting one may still hope to extract structural information about $A$
by studying the closed projections in $A^{\perp\perp}$, and this was initiated
in the work referred to above. In more recent years, this paradigm has been
successfully exploited in the context of several foundational purely operator
algebraic problems [CTh2022, CTh2023, clouatre2023pure].

The current paper is a contribution to this general program. Herein we focus
on a subclass of the projections in $A^{\perp\perp}$ that assume the role that
measurable sets satisfying $|\mu|(E)=0$ (for some natural set of measures
$\mu$) play in classical function theory. Several profound results and key
techniques in classical function theory hinge on properties of such sets, and
their relation to peak sets; see for example Chapters 9 and 10 of [rudin2008].
Our goal is to develop the noncommutative variant in order to obtain
‘quantized’ analogues of these deep facts for use in more general algebras.

We say that a projection $q\in B^{**}$ is _$A$ -null_ if $|\varphi|(q)=0$ for
every $\varphi\in A^{\perp}$. Here and throughout, given a continuous linear
function $\varphi\in B^{*}$, we denote by $|\varphi|\in B^{*}$ its modulus. It
follows that $A$-null projections automatically lie in $A^{\perp\perp}$.
Several characterizations of null projections are proved in Section 2.

When $q$ satisfies that $Aq=Bq$ as subspaces in $B^{**}$, we say that $q$ is
an _$A$ -interpolation_ projection. Note that this is a one-sided concept, in
that it is not equivalent to the equality $qA=qB$ in general (as one may see
even in the upper triangular $2\times 2$ matrices). When $B$ is commutative,
closely related notions of interpolation have long been of interest to
function theorists, even for countable sets (see for instance [AHMR2019interp]
and the references therein).

It will be helpful to highlight some of the relationship between $q$ being an
$A$-null projection, and $q$ being a (generalized) peak projection as well as
an $A$-interpolation projection. We emphasize the following result,
essentially proved in [CT2023Henkin].

Basic principle: Let $B$ be a $\mathrm{C}^{*}$-algebra and let $A\subset B$ be
a norm-closed subalgebra. If $q$ is a closed projection in $B^{**}$, then the
following statements are equivalent.

  1. (i)

$q$ is $A$-null.

  2. (ii)

$q\in A^{\perp\perp}$ and $Aq=Bq$.

###### Proof.

(ii) $\Rightarrow$(i): This is proved in [CT2023Henkin, Theorem 6.2] when $q$
is closed and $B$ unital, but the same proof works in general (it is not
necessary that $q$ be closed).

(i) $\Rightarrow$(ii): (This does not require that $A$ be a subalgebra.) That
$q\in A^{\perp\perp}$ follows from the inequality
$|\varphi(q)|^{2}\leq|\varphi|(q)\,|\varphi|(1)$ if $\varphi\in A^{\perp}$
(see Lemma 2.1 below). That $Aq=Bq$ is proved in the unital case in the first
lines of the proof of [hay2007, Proposition 3.4]. The nonunital case follows
by the same proof (which proof is given with more detail in the proof that
(ii) implies (iii) in [CT2023Henkin, Theorem 6.2]) once we know that $Aq$ is
closed.

To see this, we simply need to adapt the proof of [hay2007, Proposition 3.1]
to the case where $X=A$ and $B$ is nonunital. Consider
$B^{1}=B+{\mathbb{C}}1\subset B^{**}$. Following the argument in [hay2007], we
find a decreasing net $(e_{t})$ in $B^{1}$ converging to $q$ in the weak-$*$
topology of $B^{**}$. While $e_{t}$ may not belong to $B$, we nevertheless
have $e_{t}B\subset B$ for each index $t$, which is all that is needed in
Hay’s argument to conclude that $Aq$ is isometric to the quotient of $A$ by
some norm-closed subspace, and hence is closed. ∎

In fact, much more is true: for an $A$-null closed projection $q$, the peaking
and interpolation phenomena can be arranged to occur simultaneously in a very
strong sense, with many extra features – this is called _peak-interpolation_
(see e.g. Bishop [bishop1962]). We refer to e.g. [blecher2013, blecher2022]
for background on noncommutative peaking and interpolation, although the
latter paper focuses on a stronger two-sided version of peak-interpolation.

In light of the rigidity that the peak-interpolation phenomenon imposes on
$q$, one may guess that actually verifying that $q$ be $A$-null can be
difficult in practice. One of our main results addresses this issue, drawing
inspiration from a classical theorem of Varopoulos, which we now recall. Let
$X$ be a compact Hausdorff space and let $A\subset\mathrm{C}(X)$ be a closed
unital subalgebra. In [varopoulos1971], it is proved that in order for a
closed subset $K\subset X$ to be $A$-null, it suffices that $K$ be an
$A$-interpolation set consisting entirely of $A$-peak points.

Generalizing Varopoulos’ theorem for closed $A$-interpolation projections when
$B$ is an arbitrary $\mathrm{C}^{*}$-algebra will occupy us for the subsequent
two sections of the paper. Section 3 deals with non-commutative topological
matters that are required to properly frame our result. We seek to understand
to which extent a projection in $B^{**}$ is determined by the universally
measurable projections dominating it. This prompts us to introduce the notion
of _measurably indistinguishable_ projections, which is crucial for our
generalization. The key observation we make is Proposition 3.1, showing that
universally measurable projections are measurably indistinguishable from their
so-called _atomic parts_.

Section 4 is concerned with our noncommutative version of Varopoulos’ theorem.
For this purpose, we need the following “complete” refinement of the notion of
interpolation projection. For $\kappa>0$, we say that a projection $q$ is a
_complete $(A,\kappa)$-interpolation projection_ if, given any $n\geq 1$ and
any matrix $[b_{ij}]\in{\mathbb{M}}_{n}(B)$, there are matrices
$[a_{ij}],[a^{\prime}_{ij}]\in{\mathbb{M}}_{n}(A)$ such that

$a_{ij}q=b_{ij}q,\quad qa^{\prime}_{ij}=qb_{ij}\quad\text{ for every }1\leq
i,j\leq n$

and satisfying

$\|[a_{ij}]\|\leq\kappa\|[b_{ij}]\|,\quad\|[a^{\prime}_{ij}]\|\leq\kappa\|[b_{ij}]\|.$

When $B$ is a nuclear $\mathrm{C}^{*}$-algebra, which is the case for the main
applications we have in mind, we show in Proposition 4.8 that any closed
central $A$-interpolation projection must in fact be a complete
$(A,\kappa)$-interpolation projection for some $\kappa>0$. Before we can state
the next theorem we recall that a projection $p\in B^{**}$ is said to be _$A$
-invariant_ if $ap=pap$ for each $a\in A$. One of the main technical results
of this paper is Theorem 4.3, which goes as follows.

######  Theorem A.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
norm-closed subalgebra. Let $q\in B^{**}$ be a projection with the following
properties:

  1. (i)

$q$ is closed,

  2. (ii)

there is $\kappa>0$ for which $q$ is a complete $(A,\kappa)$-interpolation
projection,

  3. (iii)

$q$ commutes with $A$, and

  4. (iv)

the atomic part of $q$ lies in $A^{\perp\perp}$.

Then, for every $\varphi\in A^{\perp}$ and every open $A$-invariant projection
$u\in B^{**}$ dominating $q$, we have

$|\varphi|(q)\leq\kappa^{3}\sqrt{|\varphi|(1)\,|\varphi|(u-q)}.$

We interpret Theorem A as saying that for sufficiently ‘topologically regular’
interpolation projections (i.e. where such $u$ may approach $q$), the property
of being $A$-null is entirely captured by the atomic part. In Proposition 4.6,
we exhibit a non-trivial class of algebras where the atomic part of every
projection lies in $A^{\perp\perp}$. In Corollary 4.7, we identify some
natural sufficient conditions under which the conclusion of Theorem A can be
strengthened to $q$ being null. In particular, we show in Corollary 4.9 how to
recover Varopoulos’ original theorem using our results.

In recent years it has been of interest to produce peak-interpolation theorems
in the context of multiplier algebras of Hilbert function spaces; see for
instance [CD2016duality] and [DH2023]. Despite there being some obvious
conceptual overlaps between these research directions and the earlier general
noncommutative peak-interpolation theory, no explicit connection was hitherto
drawn in the literature between these two topics. In particular, the extensive
earlier noncommutative peak-interpolation machinery was not brought to bear in
proving results in [CD2016duality] and [DH2023]. In Section 5, we illustrate
how this can be done. The key observation making this possible is found in
Theorem 5.5, where we establish a bijection between null projections and
certain small sets on the unit sphere. As an application of Theorem A and of
the noncommutative peak-interpolation theory, we then give an alternative
proof for several of the results found in [DH2023]. In fact, we achieve a
sharper conclusion, since we show that peak-interpolation can be implemented
while maintaining domination by a control function, in line with Bishop’s
original theorem [bishop1962]. We call this _Bishop peak-interpolation_.

We omit the precise definitions for now, and refer the reader to the relevant
section for more detail. Let ${\mathcal{H}}$ be a Hilbert space of analytic
functions on the open unit ball ${\mathbb{B}}_{d}\subset{\mathbb{C}}^{d}$,
admitting a reproducing kernel that is unitarily invariant and regular. There
is a central projection $z\in{\mathcal{T}}({\mathcal{H}})^{**}$ and a natural
$*$-isomorphism
$\Theta:{\mathcal{T}}({\mathcal{H}})^{**}z\to\mathrm{C}({\mathbb{S}}_{d})^{**}$.

######  Theorem B.

For a closed projection $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ the following
are equivalent.

  1. (i)

$q$ is an ${\mathcal{A}}({\mathcal{H}})$-peak projection.

  2. (ii)

$q$ is ${\mathcal{A}}({\mathcal{H}})$-null.

  3. (iii)

$q$ is dominated by $z$ and there is a closed
${\mathcal{M}}({\mathcal{H}})$-totally null subset $E\subset{\mathbb{S}}_{d}$
such that $\Theta(q)=\chi_{E}$.

  4. (iv)

$q$ is dominated by $z$ and there is a closed Bishop peak-interpolation subset
$E\subset{\mathbb{S}}_{d}$ such that $\Theta(q)=\chi_{E}$.

If there exist non-empty ${\mathcal{M}}({\mathcal{H}})$-totally null sets,
then these statements are further equivalent to the following.

  1. (v)

$q$ is dominated by $z$ and there is a closed
${\mathcal{A}}({\mathcal{H}})$-interpolation set $E\subset{\mathbb{S}}_{d}$
such that $\Theta(q)=\chi_{E}$.

Moreover $\Theta$ restricts to an order preserving bijection from the
projections $q$ as above to the ${\mathcal{M}}({\mathcal{H}})$-totally null
sets $E$ (equivalently, Bishop peak-interpolation sets or
${\mathcal{A}}({\mathcal{H}})$-peak sets).

Beyond this result, our approach can be interpreted as general method that we
hope will be applicable in other contexts to prove strong peak-interpolation
results.

In Section 6, we turn to a different problem regarding $A$-null projections.
Let $\Delta$ be a norm-closed convex set of states on $B$. We let
$\operatorname{AC}(\Delta)\subset B^{*}$ denote the set of those functionals
that are absolutely continuous with respect to some element of $\Delta$. It is
known [CT2023Henkin, Theorem 3.5] that there is a unique projection
$r_{\Delta}\in B^{**}$ with the property that
$\operatorname{AC}(\Delta)^{\perp}=(1-r_{\Delta})B^{**}$. We call this
projection the _Riesz projection_ corresponding to $\Delta$. We aim to
characterize when this Riesz projection satisfies that $1-r_{\Delta}$ is
$A$-null.

More precisely, our main result relates this condition to other natural
properties inspired by classical results in function theory. We say that $A$
has the _F. &M. Riesz property_ in $B$ with respect to $\Delta$ if
$\varphi(r_{\Delta}\cdot)\in A^{\perp}$ for every $\varphi\in A^{\perp}$. This
property has been a recurring theme (under various guises) in the operator
algebras literature; see for instance [BL2007, CH2022] and the references
therein.

In addition, we say that $A$ has the _Forelli property_ in $B$ with respect to
$\Delta$ if for every projection $q\in\operatorname{AC}(\Delta)^{\perp}$ that
is the supremum of a collection of closed projections, there is a contractive
net $(a_{i})$ in $A$ such that $(qa_{i})$ converges to $0$ in the weak-$*$
topology of $B^{**}$, and $(\varphi(1-a_{i}))$ converges to $0$ for every
$\varphi\in\operatorname{AC}(\Delta)$. This technical approximation property
is directly inspired by [rudin2008, Lemma 9.5.5]. Our next main result can be
streamlined as follows.

######  Theorem C.

Assume that $\Delta$ is closed in the weak-$*$ topology of $B^{**}$. Consider
the following statements.

  1. (i)

The projection $1-r_{\Delta}$ is $A$-null.

  2. (ii)

All closed projections in $\operatorname{AC}(\Delta)^{\perp}$ are dominated by
a projection in $\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$.

  3. (iii)

$A$ has the Forelli property in $B$ with respect to $\Delta$.

  4. (iv)

$A$ has the F.&M. Riesz property in $B$ with respect to $\Delta$.

Then,

${\rm(i)}\Rightarrow{\rm(ii)}\Leftrightarrow{\rm(iii)}\Leftrightarrow{\rm(iv)}.$

The condition that $\Delta$ be weak-$*$ closed is used in proving Lemma 6.4, a
non-commutative analogue of a classical lemma due to Rainwater
[rainwater1969]. Without it, the topological nature of the Riesz projection is
not as clear. Nevertheless, many of the implications above remain valid if
$\Delta$ is merely assumed to be norm-closed (see Theorem 6.5). We apply these
findings in Section 7 to the existence of Lebesgue decompositions for operator
algebras, which is a main theme in the recent paper of the second author and
Hartz [CH2022]. In Corollary 7.4, we give a new sufficient condition for such
decompositions to exist.

In the last two sections of the paper, we aim to understand when all the
statements in Theorem B are in fact equivalent. Our first approach, used in
Section 8, consists of comparing the F.& M. Riesz property defined above with
a stronger variant of it. The key result in this direction is Corollary 8.2,
showing that the so-called _strong_ F.& M. Riesz property can be reformulated
in terms of a band property. This is in fact a consequence of a certain
rigidity phenomenon (Theorem 8.1) resulting from the F.& M. Riesz property,
and which is related to the classical Cole-Range theorem [rudin2008, Theorem
9.6.1]. This in turn is shown in Examples 6 and 7 to unify some previously
known results on the structure of Henkin functionals [CD2016duality, CH2022,
CT2023Henkin]. The property of the set of Henkin functionals forming a band
plays an important role in these sections. For example our analysis shows that
much of the function theory developed in Chapter 9 of [rudin2008], and in
particular the Cole-Range theorem, rests on two main pillars: that the Henkin
functionals form band, and the F.& M. Riesz property.

Finally, in Section 9, we tackle the equivalence question for Theorem B using
a different route, centered around the following notion. We say that $A$ has
the _Amar–Lederer property in $B$ with respect to $\Delta$_ if all closed
projections in $\operatorname{AC}(\Delta)^{\perp}$ are dominated by a closed
projection in $\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$. A very
important classical result of Amar–Lederer [AL1971] shows that
$H^{\infty}({\mathbb{D}})$ satisfies this property inside
$L^{\infty}({\mathbb{T}})$ with respect to the state of integration against
arclength measure: speaking slightly loosely they show that any Lebesgue null
Borel set is contained in a (closed) Lebesgue null peak set. On the other
hand, the statements in Theorem B fail to be equivalent in this case (Example
8). The rest of the section is devoted to proving Theorem 9.4, where we
contribute to the theory of maximal subdiagonal algebras. Indeed, we refine
known results from [ueda2009, BL2018] using some new noncommutative measure
theory, and show that such algebras do satisfy the full Amar and Lederer
property for all null closed (indeed ‘regular’) projections, and hence they
also have the Forelli property.

We close this introduction by mentioning some notation and background facts.
An operator algebra is a norm closed algebra of operators on a Hilbert space,
or equivalently a closed subalgebra of a $\mathrm{C}^{*}$-algebra. We refer to
[BLM2004] for their theory. We will use silently the fact from basic
functional analysis that $X^{\perp\perp}$ is the weak* closure in $Y^{**}$ of
a subspace $X\subset Y$, and is isometric to $X^{**}$. For us a projection is
always an orthogonal projection. The bidual of an operator algebra $A$ is an
operator algebra, indeed is a von Neumann algebra if $A$ is selfadjoint. We
write $\chi_{E}$ for the characteristic function of a set $E$. In the case
that $B=\mathrm{C}(K)$, and $E$ is an open or closed set in $K$, the
projection $q=\chi_{E}$ may be viewed as an element of $\mathrm{C}(K)^{**}$ in
a natural way since $\mathrm{C}(K)^{*}$ is a certain space of measures on $K$.
Thus if $B=\mathrm{C}(K)$ the open or closed projections are precisely the
characteristic functions of open or closed sets. Open projections arise
naturally in functional analysis. For example, they come naturally out of the
‘spectral theorem/functional calculus’: the spectral projections in $B^{**}$
of a self-adjoint operator $a\in B$ corresponding to open sets in the spectrum
of $T$ are open projections. The nullspace of a normal state on a von Neumann
algebra is closed under ‘joins’ and ‘meets’ of projections.

##  2\. Null projections

Let $B$ be a $\mathrm{C}^{*}$-algebra and let $M\subset B$ be a subspace.
Recall that a projection $p\in B^{**}$ is _$M$ -null_ if $|\varphi|(p)=0$ for
every $\varphi\in B^{*}$ annihilating $M$. In this section, we collect some
general results on null projections. Before proceeding, we record an
elementary inequality that will be used numerous times throughout the paper.

######  Lemma 2.1.

Let $B$ be a $\mathrm{C}^{*}$-algebra and let $\varphi\in B^{*}$. Let $p\in
B^{**}$ be a projection. Then, for every $x\in B^{**}$ we have
$|\varphi(px)|^{2}\leq\|\varphi\|\|x\|^{2}|\varphi|(p).$

###### Proof.

Choose a partial isometry $v\in B^{**}$ such that $\varphi=|\varphi|(\cdot
v)$. Thus, by the Schwarz inequality we obtain that

$|\varphi(px)|^{2}=||\varphi|(pxv)|^{2}\leq\|\varphi\||\varphi|(pxv(pxv)^{*})\leq\|\varphi\|\,\|x\|^{2}\,|\varphi|(p)$

for $x\in B^{**}$. ∎

Recall that a closed projection $p\in B^{**}$ is _compact_ if there is a
contraction $b\in B$ such that $p=pb$. When $B$ is unital, this is clearly the
same notion as that of being closed. We note that if $B$ is a nonunital
$\mathrm{C}^{*}$-algebra and $A\subset B$ be a norm-closed subalgebra, then
the generalized $A$-peak projections are the compact projections that happen
to lie in $A^{\perp\perp}$. See [BR2013, BN2012] for this generalization of
Glicksberg’s theorem. Thus in this case our “Basic principle” stated in the
introduction informs us that compact $A$-null projections are exactly the
generalized $A$-peak projections which are also interpolation projections.

The next development hinges on the following basic fact.

######  Lemma 2.2.

Let $B$ be a $\mathrm{C}^{*}$-algebra. Then, the compact projections in
$B^{**}$ span a weak-$*$ dense subspace.

###### Proof.

Let $D\subset B^{**}$ denote the subspace generated by all compact projections
in $B^{**}$. Let $\varphi\in B^{*}$ be a functional annihilating $D$. We claim
that $\varphi=0$. To see this, it suffices to fix a positive element $b\in B$
and to show that $\varphi(b)=0$. Let $\Omega\subset{\mathbb{R}}$ denote the
spectrum of $b$. We recall that $\mathrm{C}_{0}(\Omega\setminus\\{0\\})$ is
$*$-isomorphic to the $\mathrm{C}^{*}$-algebra generated by $b$. Then there
exists a regular Borel measure $\mu$ on $\Omega\setminus\\{0\\}$ such that

(1) $\varphi(f(b))=\int_{\Omega}fd\mu$

for every $f\in\mathrm{C}_{0}(\Omega)$ with $f(0)=0$. If
$K\subset\Omega\setminus\\{0\\}$ is compact then $\chi_{K}(b)\in B^{**}$ by
functional calculus, and $\chi_{K}(b)$ is closed with respect to the
unitization of $B$, hence it is a compact projection in $B^{**}$[AAP1989,
Lemma 2.4]. Hence, $\chi_{K}(b)\in D$ and $\varphi(\chi_{K}(b))=0$. On the
other hand, by a dominated convergence argument applied to a sequence of
continuous functions decreasing to $\chi_{K}$ and vanishing at $0$, we find by
virtue of (1) that $\varphi(\chi_{K}(b))=\mu(K)$ so that $\mu(K)=0$. By
regularity of the measure $\mu$, we conclude that $\mu=0$, and thus
$\varphi=0$. ∎

As motivation for what is to come, we give a simple criterion for the
agreement between a $\mathrm{C}^{*}$-algebra and one of its subspaces. For
subalgebras of commutative $\mathrm{C}^{*}$-algebras, one may use such
considerations to recover classical characterizations of $\mathrm{C}(K)$ among
its subalgebras, such as one due to Bade–Curtis [BC1966, Theorem B].

######  Proposition 2.3.

Let $B$ be a $\mathrm{C}^{*}$-algebra and let $M\subset B$ be a norm-closed
subspace. Then, the following statements are equivalent.

  1. (i)

$M=B$.

  2. (ii)

The unit of $B^{**}$ is $M$-null.

  3. (iii)

Any compact projection in $B^{**}$ lies in $M^{\perp\perp}$.

###### Proof.

(i)$\Rightarrow$(ii): By assumption, we see that $M^{\perp}=\\{0\\}$, so that,
vacuously, the unit of $B^{**}$ is $M$-null.

(ii) $\Rightarrow$(iii): Let $\varphi\in M^{\perp}$ and $p\in B^{**}$ be a
compact projection. Let $e\in B^{**}$ denote the unit. Then, $p\leq e$ so that
$|\varphi|(p)\leq|\varphi|(e)=0$. This shows that $p$ is $M$-null, and in
particular $p\in M^{\perp\perp}$ by Lemma 2.1.

(iii)$\Rightarrow$(i): The assumption combined with Lemma 2.2 implies that
$M^{\perp\perp}=B^{**}$. In turn, by the Hahn–Banach theorem, we find $M=B$
since $M$ is norm-closed. ∎

A natural question that arises from Proposition 2.3 is whether there is a
similar characterization of nullity for proper projections, as opposed to the
unit of $B^{**}$ in that result. As was communicated to us by Alexander Izzo,
this is indeed the case for subalgebras of commutative
$\mathrm{C}^{*}$-algebras, at least for Borel projections – this conversation
with Izzo inspired several results in this section; see also the remark at the
end of Section 6.

More precisely, let $X$ be a compact Hausdorff space and let
$A\subset\mathrm{C}(X)$ be a unital norm-closed subalgebra. Let $E\subset X$
be a Borel measurable subset, and denote by $\chi_{E}\in\mathrm{C}(X)^{**}$
its characteristic function. Then, $\chi_{E}$ is $A$-null if and only if
$\chi_{K}\in A^{\perp\perp}$ for every closed subset $K\subset E$. Our next
goal is to establish such a characterization in the non-commutative context.

######  Proposition 2.4.

Let $B$ be a $\mathrm{C}^{*}$-algebra and let $M\subset B$ be a norm-closed
subspace. Let $p\in B^{**}$ be a projection. Then, the following statements
are equivalent.

  1. (i)

$pB^{**}\subset M^{\perp\perp}$.

  2. (ii)

For every compact projection $q\in B^{**}$, the element $pq$ lies in
$M^{\perp\perp}$.

  3. (iii)

The projection $p$ is $M$-null.

###### Proof.

(i) $\Rightarrow$ (ii): This is trivial.

(ii) $\Rightarrow$ (iii): Fix $\varphi\in M^{\perp}$. By assumption,
$\varphi(pq)=0$ for every compact projection $q\in B^{**}$. By Lemma 2.2, it
follows that $\varphi(px)=0$ for every $x\in B^{**}$. In particular,
$|\varphi|(p)=0$.

(iii) $\Rightarrow$ (i): Let $\varphi\in M^{\perp}$. Then $|\varphi|(p)=0$, so
that $\varphi(px)=0$ for each $x\in B^{**}$ by Lemma 2.1. We conclude that
$pB^{**}\subset M^{\perp}$. ∎

When $B$ is unital and commutative and $p$ is closed, statement (ii) above is
equivalent to the stipulation that every closed projection dominated by $p$
lies in $M^{\perp\perp}$. Next, we give an alternate version of Proposition
2.4 that is more aligned with this formulation. For this purpose, we require
some ideas from [blecher2022].

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $q\in B^{**}$ be a closed
projection. Denote by $B_{q}\subset B$ the $\mathrm{C}^{*}$-subalgebra of
elements in $B$ commuting with $q$. We say that a projection $r\in
B_{q}^{\perp\perp}$ is _closed with respect to $B_{q}$_ if there is a net
$(b_{i})$ of positive contractions in $B_{q}$ decreasing to $r$ in the
weak-$*$ topology of $B^{**}$. Equivalently, using the weak-$*$ homeomorphic
$*$-isomorphism $B_{q}^{**}\cong B_{q}^{\perp\perp}$, this means that $r$ is
closed when viewed as an element of $B_{q}^{**}$. It is easily verified that
$q$ itself lies in $B_{q}^{\perp\perp}$, and it is closed with respect to
$B_{q}$ (see for instance [blecher2022, Lemma 2.1]). Indeed a projection $r\in
B_{q}^{\perp\perp}$ is closed with respect to $B_{q}$ if and only if it is
closed with respect to $B$.

Next, let $A\subset B$ be a norm-closed subalgebra and put $A_{q}=A\cap
B_{q}$. Then, $A_{q}$ may be viewed as a subalgebra of both $B^{**}$ and
$B_{q}^{**}$. Via the isomorphism $B_{q}^{**}\cong B_{q}^{\perp\perp}$, the
double annihilator of $A_{q}$ inside of $B_{q}^{**}$ and its double
annihilator inside of $B^{**}$ are identified with one another. In what
follows, we will reserve the notation $A_{q}^{\perp\perp}$ for the double
annihilator of $A_{q}$ inside of $B^{**}$.

######  Proposition 2.5.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra, and let $A\subset B$ be a unital
norm-closed subalgebra. Let $q\in B^{**}$ be a closed projection lying in
$A^{\perp\perp}$. Then, the following statements are equivalent.

  1. (i)

Every closed subprojection $p$ of $q$ in $B_{q}^{\perp\perp}$ is a generalized
$A_{q}$-peak projection (that is, $p\in A_{q}^{\perp\perp}$).

  2. (ii)

$q$ is $A_{q}$-null with respect to $B_{q}$.

  3. (iii)

$q$ is $A_{q}$-interpolating with respect to $B_{q}$.

###### Proof.

By Lemma 2.1 in [blecher2022] we automatically have $q\in A_{q}^{\perp\perp}$.
Then the equivalence of (ii) and (iii) follows from the “Basic principle”
stated in the introduction.

(i) $\Rightarrow$(ii): Let $\varphi:B_{q}\to{\mathbb{C}}$ be a continuous
linear functional annihilating $A_{q}$. Let $v\in B_{q}^{**}$ be a partial
isometry such that $|\varphi|=\varphi(\cdot v)$. The goal is to show that
$\varphi(qv)=0$. By virtue of Lemma 2.2, it suffices to show that
$\varphi(qp)=0$ for every closed projection $p\in B_{q}^{**}$. Any such
projection $p$ necessarily commutes with $q$ in $B_{q}^{**}$, so that $qp$ is
closed in $B_{q}^{**}$ [akemann1969, Theorem II.7] and is dominated by $q$. In
turn, viewing $qp$ as a closed subprojection of $q$ in $B_{q}^{\perp\perp}$,
the assumption gives $qp\in A_{q}^{\perp\perp}$, so that as a projection in
$B_{q}^{**}$, $qp$ lies in the double annihilator of $A_{q}$. In particular,
$\varphi(qp)=0$ as desired.

(ii)$\Rightarrow$(i): Let $p\in B^{**}$ be any projection lying in
$B_{q}^{\perp\perp}$ and dominated by $q$. Then, $p$ may be viewed as a
projection in $B_{q}^{**}$ dominated by $q$. Our assumption then implies that
$|\varphi|(p)=0$ for every $\varphi\in B_{q}^{*}$ annihilating $A_{q}$. In
particular, by virtue of Lemma 2.1, we infer that $\varphi(p)=0$ for all such
functionals $\varphi$, and hence $p$ lies in the double annihilator of $A_{q}$
inside of $B_{q}^{**}$. In turn, when viewed as a projection in $B^{**}$, this
means that $p$ lies in $A_{q}^{\perp\perp}$. ∎

##  3\. Measurably indistinguishable projections

This short section contains a preliminary discussion on some non-commutative
topological notions for projections. These are used in the subsequent section.

We start with a motivating discussion. Given a topological space $X$
satisfying the $T_{1}$ separation axiom and distinct subsets $E,F\subset X$,
there always exists an open subset $U\subset X$ containing $E$ but not
containing $F$. In Akemann’s non-commutative topology for projections, more
diverse phenomena can arise, even in commutative contexts.

######  Example 1.

Let ${\mathbb{T}}\subset{\mathbb{C}}$ denote the unit circle and let
$B=\mathrm{C}({\mathbb{T}})$. For each $t\in{\mathbb{T}}$, consider the closed
projection $p_{t}=\chi_{\\{t\\}}\in B^{**}$. Let $\lambda\in B^{*}$ denote the
state of integration against arc length measure. Let $e\in B^{**}$ be the
projection satisfying

$(1-e)B^{**}=\\{x\in B^{**}:\lambda(x^{*}x)=0\\}.$

Since $\\{t\\}$ has measure zero, we see that $\lambda(p_{t})=0$ so that
$p_{t}\leq 1-e$ for each $t\in{\mathbb{T}}$, and thus
$\bigvee_{t\in{\mathbb{T}}}p_{t}\leq 1-e$. Fix a Borel measurable subset
$E\subset{\mathbb{T}}$ with non-zero measure. We clearly have $\bigvee_{t\in
E}p_{t}\leq\chi_{E}$. On the other hand, $\chi_{E}(\lambda)>0$ so $\chi_{E}$
is not dominated by $1-e$, and in particular, $\chi_{E}\neq\bigvee_{t\in
E}p_{t}$.

Nevertheless, the projections $\chi_{E}$ and $\bigvee_{t\in E}p_{t}$ are
nearly identical: their differences cannot be detected by means of certain
Borel projections. More precisely, let $F\subset{\mathbb{T}}$ be a Borel
subset. Assume that $\bigvee_{t\in E}p_{t}\leq\chi_{F}$, so that
$\chi_{\\{t\\}}\leq\chi_{F}$ and $t\in F$ for each $t\in E$. This means that
$E\subset F$, or $\chi_{E}\leq\chi_{F}$. In other words, while $\bigvee_{t\in
E}p_{t}$ and $\chi_{E}$ are distinct projections, they are dominated by
precisely the same projections arising from Borel subsets. ∎

Let $B$ be a $\mathrm{C}^{*}$-algebra. By considering the usual universal
representation of $B$, we find a Hilbert space $H_{u}$ such that
$B^{**}\subset B(H_{u})$. We let $(B_{\text{sa}})^{m}\subset B^{**}$ denote
the set of elements in $B^{**}$ that can be obtained as the increasing limit
in the SOT of $B(H_{u})$ of a self-adjoint net from $B$. A self-adjoint
element $x\in B^{**}$ is said to be _universally measurable_ if, for every
$\varepsilon>0$ and state $\varphi$ on $B$, there are elements
$y,z\in(B_{\text{sa}})^{m}$ such that $-y\leq x\leq z$ and
$\varphi(y+z)<\varepsilon$.

Projections in $B^{**}$ that are either open or closed are universally
measurable [pedersen1979book, Proposition 3.11.9]. More generally, the class
of universally measurable projections also contains the Borel projections in
the sense of [combes1970]; see [brown1970] for more detail.

Let $p,q\in B^{**}$ be two projections. We say that $p$ and $q$ are
_measurably indistinguishable_ if they are dominated by the same universally
measurable projections in $B^{**}$. We will be concerned with the issue of
measurable indistinguishability between a projection and the supremum of all
minimal projections that it dominates. This is best framed using the following
device.

Let $z_{\operatorname{at}}\in B^{**}$ denote the supremum of all minimal
projections in $B^{**}$. It is well known that $z_{\operatorname{at}}$ is
central: indeed, it coincides with the support projection for the unique
weak-$*$ continuous extension to $B^{**}$ of the _atomic representation_ of
$B$ [pedersen1979book, Paragraph 4.3.7]. In particular, by virtue of
[pedersen1979book, Lemma 4.3.8], we see that there is a collection
$\\{H_{\lambda}:\lambda\in\Lambda\\}$ of Hilbert spaces such that
$z_{\operatorname{at}}B^{**}\cong\prod_{\lambda\in\Lambda}B(H_{\lambda})$.
Consequently, if $p\in B^{**}$ is a projection, we have

(2) $pz_{\operatorname{at}}=\bigvee\\{r:r\text{ minimal projection in
}B^{**},r\leq p\\}.$

Henceforth, we will refer to $pz_{\operatorname{at}}$ as the _atomic part_ of
$p$.

######  Proposition 3.1.

Universally measurable projections in $B^{**}$ are measurably
indistinguishable from their atomic parts.

###### Proof.

Let $p,e\in B^{**}$ be universally measurable projections with $e\geq
pz_{\operatorname{at}}$. Since $z_{\operatorname{at}}$ is central, we find
$(e-p)z_{\operatorname{at}}\geq 0$. In turn, because $e-p$ is universally
measurable [pedersen1979book, Proposition 4.3.13], we infer that $e\geq p$ by
virtue of [pedersen1979book, Theorem 4.3.15]. ∎

Let us further analyze the case of closed projections, as this is what we will
focus on in the next section. The previous proposition can be interpreted as
saying that, roughly speaking, a closed projection in $B^{**}$ essentially
coincides with the supremum of all the minimal projections that it dominates.
Of course, as illustrated in Example 1, this is not strictly true.
Nevertheless, the following does hold.

######  Corollary 3.2.

Let $q\in B^{**}$ be a closed projection. Let $C\subset B^{**}$ denote the
collection of all closed projections dominating the same minimal projections
as $q$ does. Then, $q=\bigwedge_{r\in C}r$.

###### Proof.

Since $z_{\operatorname{at}}$ is central, we have $q\geq
qz_{\operatorname{at}}$, so that $q\in C$ and $q\geq\bigwedge_{r\in C}r$.
Conversely, fix $r\in C$ and an open projection $u\in B^{**}$ with $u\geq r$.
Then, $u\geq qz_{\operatorname{at}}$ by (2), so by Proposition 3.1, it follows
that $u\geq q$. Since this holds for every such open projection $u$, applying
[hay2007, Proposition 2.3], we find that $q\leq r$. We conclude that
$q\leq\bigwedge_{r\in C}r$, so in fact $q=\bigwedge_{r\in C}r$ as desired. ∎

##  4\. The non-commutative Varopoulos phenomenon

We begin by revisiting Example 1 to illustrate that a closed projection may
have a null atomic part without being null itself, despite both projections
being measurably indistinguishable by Proposition 3.1.

######  Example 2.

Let ${\mathbb{T}}\subset{\mathbb{C}}$ denote the unit circle, let
$B=\mathrm{C}({\mathbb{T}})$. For each $t\in{\mathbb{T}}$, consider the closed
projection $p_{t}=\chi_{\\{t\\}}\in B^{**}$. Let $\lambda\in B^{*}$ denote the
state of integration against arc length measure. Let $K\subset{\mathbb{T}}$ be
a closed subset with positive measure. The atomic part of the closed
projection $\chi_{K}\in B^{**}$ is easily seen to be $\bigvee_{t\in K}p_{t}$
by (2).

Next, let $A\subset B$ denote the disc algebra, i.e. the norm-closed
subalgebra generated by analytic polynomials. It is well known that each point
of ${\mathbb{T}}$ is an $A$-peak point, so that $p_{t}$ is $A$-null for each
$t\in{\mathbb{T}}$. It follows that $\bigvee_{t\in E}p_{t}$ is also $A$-null.
On the other hand, $\lambda(\chi_{K})>0$, and hence $\chi_{K}$ is not an
$A$-null [rudin2008, Theorem 10.1.2] and not in $A^{\perp\perp}$, so that $K$
is not a peak set. ∎

The goal of this section is to exhibit conditions under which such a pathology
cannot occur. This will be accomplished by proving a non-commutative analogue
of Varopoulos’ theorem [rudin2008, Theorem 10.2.2]. First, we record a
topological preliminary.

######  Lemma 4.1.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $q\in B^{**}$ be a closed
projection. Let $\\{u_{\lambda}:\lambda\in\Lambda\\}$ be a collection of open
projections in $B^{**}$ that commute with $q$ and satisfy
$q\leq\bigvee_{\lambda\in\Lambda}u_{\lambda}$. Then, there is a finite set
$F\subset\Lambda$ such that $q\leq\bigvee_{\lambda\in F}u_{\lambda}$.

###### Proof.

By assumption, we see that
$q\wedge\left(\bigwedge_{\lambda\in\Lambda}(1-u_{\lambda})\right)=0$. Invoking
[akemann1969, Proposition II.10], we find a finite subset $F\subset\Lambda$
such that $q\wedge\left(\bigwedge_{\lambda\in F}(1-u_{\lambda})\right)=0$ or
$q\wedge\left(1-\bigvee_{\lambda\in F}u_{\lambda}\right)=0$. Equivalently,
using the commutation assumption, we see that $q\leq\bigvee_{\lambda\in
F}u_{\lambda}$ as desired. ∎

The previous result is stated to hold with no requirement of commutativity in
[hay2007], but this is not valid in general. Thankfully, the main claims of
that paper appear to be unaffected by this.

We also need the following technical fact. Given a vector space $X$, we denote
by ${\mathbb{M}}_{m,n}(X)$ the space of $m\times n$ matrices with entries in
$X$. When $m=n$, we simply write ${\mathbb{M}}_{n}(X)$.

######  Lemma 4.2.

Let $B$ be a $\mathrm{C}^{*}$-algebra, let $q\in B^{**}$ be a closed
projection and let $n\geq 1$. Let $R\in{\mathbb{M}}_{1,n}(qB^{**})$ and
$C\in{\mathbb{M}}_{n,1}(B^{**}q)$. Fix $\Delta\in{\mathbb{M}}_{n}(B)$, a
functional $\varphi\in B^{*}$ and a number $\varepsilon>0$. Then, there are
$S\in{\mathbb{M}}_{1,n}(qB)$, $T\in{\mathbb{M}}_{n,1}(Bq)$ such that
$\|S\|\leq\|R\|$,$\|T\|\leq\|C\|$ and $|\varphi(R\Delta C-S\Delta
T)|<\varepsilon.$

###### Proof.

Let $J=B\cap(1-q)B^{**}$. Consider the map $\Phi:B/J\to qB$ defined as

$\Phi(b+J)=qb,\quad b\in B.$

By [hay2007, Proposition 3.1], it follows that $\Phi$ is a completely
isometric isomorphism. Hence, $\Phi^{**}:(B/J)^{**}\to(qB)^{**}$ is a
completely isometric weak-$*$ homeomorphism. In turn, by [BLM2004, Paragraph
1.4.4], there is a completely isometric weak-$*$ homeomorphism
$\Psi:(B/J)^{**}\to B^{**}/J^{\perp\perp}$ such that

$\Psi(b+J)=b+J^{\perp\perp},\quad b\in B.$

Next, since $q$ is closed, we obtain that $J^{\perp\perp}=(1-q)B^{**}$. Hence,
the map $\Theta:B^{**}/J^{\perp\perp}\to qB^{**}$ defined as

$\Theta(x+J^{\perp\perp})=qx,\quad x\in B^{**}$

is also a completely isometric weak-$*$ homeomorphism. Therefore, we obtain a
completely isometric weak-$*$ homeomorphism
$\Omega=\Theta\circ\Psi\circ(\Phi^{**})^{-1}:(qB)^{**}\to qB^{**}$ acting as
the inclusion on $qB$.

Invoke now [BLM2004, Theorem 1.4.11] to find an isometric isomorphism

$\Gamma:{\mathbb{M}}_{1,n}(qB)^{**}\to{\mathbb{M}}_{1,n}((qB)^{**})$

acting as the inclusion on ${\mathbb{M}}_{1,n}(qB)$. Then, the map
$\Xi=\Omega^{(n)}\circ\Gamma:{\mathbb{M}}_{1,n}(qB)^{**}\to{\mathbb{M}}_{1,n}(qB^{**})$
is an isometric weak-$*$ homeomorphism acting as the inclusion on
${\mathbb{M}}_{1,n}(qB)$. Consider the weak-$*$ continuous linear functional
$\psi$ on ${\mathbb{M}}_{1,n}(qB)^{**}$ defined as

$\psi(X)=\varphi(\Xi(X)\Delta C),\quad X\in{\mathbb{M}}_{1,n}(qB)^{**}.$

By Goldstine’s lemma, we can find $S_{0}\in{\mathbb{M}}_{1,n}(qB)$ with
$\|S_{0}\|\leq\|\Xi^{-1}(R)\|=\|R\|$ such that

$|\psi(S_{0}-\Xi^{-1}(R))|<\varepsilon/2.$

Put $S=\Xi(S_{0})\in{\mathbb{M}}_{1,n}(qB)$, which then satisfies
$\|S\|\leq\|R\|$ and

(3) $|\varphi(S\Delta C-R\Delta C)|<\varepsilon/2.$

To find $T$, we argue in a similar fashion. First, proceed as above to find an
isometric weak-$*$ homeomorphism
$\Xi^{\prime}:{\mathbb{M}}_{n,1}(Bq)^{**}\to{\mathbb{M}}_{n,1}(B^{**}q)$
acting as the inclusion on ${\mathbb{M}}_{n,1}(Bq)$. Consider the weak-$*$
continuous linear functional $\psi^{\prime}$ on ${\mathbb{M}}_{n,1}(Bq)^{**}$
defined as

$\psi^{\prime}(X)=\varphi(S\Delta\Xi^{\prime}(X)),\quad
X\in{\mathbb{M}}_{n,1}(Bq)^{**}.$

By Goldstine’s lemma, we can find $T_{0}\in{\mathbb{M}}_{1,n}(Bq)$ with
$\|T_{0}\|\leq\|(\Xi^{\prime})^{-1}C\|=\|C\|$ such that

$|\psi^{\prime}(T_{0}-(\Xi^{\prime})^{-1}(C))|<\varepsilon/2.$

Put $T=\Xi^{\prime}(T_{0})\in{\mathbb{M}}_{1,n}(Bq)$, which then satisfies
$\|T\|\leq\|C\|$ and

(4) $|\varphi(S\Delta T-S\Delta C)|<\varepsilon/2.$

Finally, combining (3) and (4), we find

$|\varphi(R\Delta C-S\Delta T)|<\varepsilon$

as desired. ∎

Let $B$ be a $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a subalgebra. A
projection $e\in B^{**}$ is said to be _$A$ -invariant_ if $ae=eae$ for each
$a\in A$. Further, $e$ is said to be _$A$ -coinvariant_ if $1-e$ is
$A$-invariant. We can now state and prove one of the main results of the
paper.

######  Theorem 4.3.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
norm-closed subalgebra. Let $q\in B^{**}$ be a projection with the following
properties:

  1. (i)

$q$ is closed,

  2. (ii)

there is $\kappa>0$ for which $q$ is a complete $(A,\kappa)$-interpolation
projection,

  3. (iii)

$q$ commutes with $A$, and

  4. (iv)

the atomic part of $q$ lies in $A^{\perp\perp}$.

Then, for every $\varphi\in A^{\perp}$ and every open $A$-invariant projection
$u\in B^{**}$ dominating $q$, we have

$|\varphi|(q)\leq\kappa^{3}\sqrt{|\varphi|(1)\,|\varphi|(u-q)}.$

###### Proof.

Let $0<\varepsilon<1$ and let $\varphi\in A^{\perp}$. We may assume without
loss of generality that $\|\varphi\|=1$, so that $|\varphi|$ is a state.
Throughout, we put $\tau=|\varphi|$. There is a partial isometry $y\in B^{**}$
such that $\tau=\varphi(\cdot y)$ and $\varphi=\tau(\cdot y^{*})$. By
Goldstine’s lemma, there is $t\in B$ such that $\|t\|\leq 1$ and $|\varphi(qy-
qt)|<\varepsilon.$ Because $q$ is a complete $(A,\kappa)$-interpolation
projection, there is $x\in A$ such that $qx=qt$ and $\|x\|\leq\kappa$. In
particular, we have

(5) $|\tau(q)|=|\varphi(qy)|<\varepsilon+|\varphi(qx)|.$

Let $\\{p_{\lambda}:\lambda\in\Lambda\\}$ be the set of minimal projections in
$B^{**}$ dominated by $q$, so that
$q_{\operatorname{at}}=\bigvee_{\lambda\in\Lambda}p_{\lambda}$ is the atomic
part of $q$ by (2). It follows from [akemann1969, Proposition II.4] that each
$p_{\lambda}$ is closed. Further, since $qA=qB$ we necessarily have
$q_{\operatorname{at}}A=q_{\operatorname{at}}B$. Also,
$q_{\operatorname{at}}\in A^{\perp\perp}$ by assumption, so the “Basic
principle” from the introduction implies that $q_{\operatorname{at}}$ is
$A$-null, and thus so is each $p_{\lambda}$. In particular, we have
$p_{\lambda}\in A^{\perp\perp}$ for every $\lambda\in\Lambda$.

Let $u\in B^{**}$ be an open $A$-invariant projection dominating $q$. In
particular, $u$ dominates the atomic part of $q$, so that $u\geq p_{\lambda}$
for every $\lambda$. For each $\lambda$, we may thus apply [BN2012, Theorem
2.1] to find a contraction $a_{\lambda}\in A$ such that
$a_{\lambda}p_{\lambda}=p_{\lambda}$ and

(6) $\|(1-u)a_{\lambda}\|<\varepsilon.$

Consider the open projection
$f_{\lambda}=\chi_{[0,\varepsilon)}(|1-a_{\lambda}|)\in B^{**}$. Note that $q$
commutes with $A$, and hence with $f_{\lambda}$ as well. Since
$\|a_{\lambda}\|=1$ and $a_{\lambda}p_{\lambda}=p_{\lambda}$, we infer
$|1-a_{\lambda}|p_{\lambda}=0$ and thus

$(1-f_{\lambda})p_{\lambda}=\chi_{[\varepsilon,1]}(|1-a_{\lambda}|)p_{\lambda}=0.$

Therefore, $p_{\lambda}\leq f_{\lambda}$. It follows that the open projection
$\bigvee_{\lambda}f_{\lambda}$ dominates $\bigvee_{\lambda}p_{\lambda}$, and
hence it also dominates $q$ by Proposition 3.1. We may apply Lemma 4.1 to find
finitely many indices $\lambda_{1},\ldots,\lambda_{n}$ such that $q\leq
f_{\lambda_{1}}\vee\ldots\vee f_{\lambda_{n}}$.

Put $e_{1}=f_{\lambda_{1}}$, and for $2\leq j\leq n$, let
$e_{j}=\bigvee_{i=1}^{j}f_{\lambda_{i}}-\bigvee_{i=1}^{j-1}f_{\lambda_{i}}$.
It follows that the collection $\\{e_{1},\ldots,e_{n}\\}$ is pairwise
orthogonal and $\sum_{j=1}^{n}e_{n}=\bigvee_{j=1}^{n}f_{\lambda_{j}}$.
Invoking [takesaki2002, Proposition V.1.6], we find for each $2\leq j\leq n$ a
partial isometry $v_{j}\in B^{**}$ such that $v_{j}^{*}v_{j}=e_{j}$ and
$v_{j}v_{j}^{*}=e^{\prime}_{j}$, where
$e_{j}^{\prime}=f_{\lambda_{j}}-\left(\bigvee_{i=1}^{j-1}f_{\lambda_{i}}\right)\wedge
f_{\lambda_{j}}.$ For convenience, put $v_{1}=1$ and $e_{1}^{\prime}=e_{1}$.
We compute, for $1\leq j\leq n$, that

$\displaystyle v_{j}^{*}a_{\lambda_{j}}v_{j}e_{j}-e_{j}$
$\displaystyle=v_{j}^{*}a_{\lambda_{j}}v_{j}v_{j}^{*}v_{j}-v_{j}^{*}v_{j}v_{j}^{*}v_{j}$
$\displaystyle=v_{j}^{*}(a_{\lambda_{j}}e^{\prime}_{j}-e^{\prime}_{j})v_{j}$

whence

$\|v_{j}^{*}a_{\lambda_{j}}v_{j}e_{j}-e_{j}\|\leq\|(a_{\lambda_{j}}-1)e_{j}^{\prime}\|\leq\||1-a_{\lambda_{j}}|f_{\lambda_{j}}\|$

since $e^{\prime}_{j}\leq f_{\lambda_{j}}$. We conclude, for each $1\leq j\leq
n$, that

(7) $\|v_{j}^{*}a_{\lambda_{j}}v_{j}e_{j}-e_{j}\|\leq\varepsilon.$

Consider
$V=\begin{bmatrix}qe_{1}v_{1}^{*}&\ldots&qe_{n}v_{n}^{*}\end{bmatrix}$ and
$E=\begin{bmatrix}qe_{1}&\ldots&qe_{n}\end{bmatrix}$ as elements in
${\mathbb{M}}_{1,n}(qB^{**})$. We find

$VV^{*}=\sum_{j=1}^{n}qe_{j}v_{j}^{*}v_{j}e_{j}q=qEE^{*}q$

and

$EE^{*}=\sum_{j=1}^{n}qe_{j}q=q\left(\bigvee_{j=1}^{n}f_{\lambda_{j}}\right)q=q$

whence $\|V\|=\|E\|=1$. If we set
$\Delta=\operatorname{diag}(a_{\lambda_{1}},\ldots,a_{\lambda_{n}})$, then

$\displaystyle V\Delta V^{*}-q$
$\displaystyle=\sum_{j=1}^{n}qe_{j}v_{j}^{*}a_{\lambda_{j}}v_{j}e_{j}q-\sum_{j=1}^{n}qe_{j}q$
$\displaystyle=\sum_{j=1}^{n}qe_{j}(v_{j}^{*}a_{\lambda_{j}}v_{j}e_{j}-e_{j})e_{j}q$
$\displaystyle=E\operatorname{diag}(v_{j}^{*}a_{\lambda_{j}}v_{j}e_{j}-e_{j})E^{*}.$

By (7) we infer

(8) $\|V\Delta V^{*}-q\|\leq\varepsilon.$

By Lemma 4.2, we may now choose $s_{1},\ldots,s_{n}\in B$ and
$t_{1},\ldots,t_{n}\in B$ such that, upon setting

$S=\begin{bmatrix}s_{1}&\ldots&s_{n}\end{bmatrix}\quad\text{and}\quad
T=\begin{bmatrix}t_{1}\\\ \vdots\\\ t_{n}\end{bmatrix}$

we find

$\|qS\|\leq\|V\|=1\quad\text{and}\quad\|Tq\|\leq\|V^{*}\|=1$

along with

(9) $\left|\tau\left((V\Delta V^{*}-qS\Delta
Tq)xy^{*}\right)\right|<\varepsilon.$

Since $q$ is a complete $(A,\kappa)$-interpolation projection, we can find
$\alpha_{1},\ldots,\alpha_{n}\in A$ along with $\beta_{1},\ldots,\beta_{n}\in
A$ such that, if we put

$R=\begin{bmatrix}\alpha_{1}&\ldots&\alpha_{n}\end{bmatrix}\quad\text{and}\quad
C=\begin{bmatrix}\beta_{1}\\\ \vdots\\\ \beta_{n}\end{bmatrix},$

then

(10) $qS=qR\quad\text{and}\quad Tq=Cq$

while

$\|R\|\leq\kappa\|qS\|\leq\kappa\quad\text{and}\quad\|C\|\leq\kappa\|Tq\|\leq\kappa.$

We define $d=R\Delta C=\sum_{i=1}^{n}\alpha_{i}a_{\lambda_{i}}\beta_{i}\in A$
and note that

(11) $\|d\|\leq\|R\|\|\Delta\|\|C\|\leq\kappa^{2}.$

Since $u$ is $A$-invariant, we find
$(1-u)d=(1-u)\sum_{j=1}\alpha_{j}(1-u)a_{\lambda_{j}}\beta_{j}$ so

$\|(1-u)d\|\leq\|R\|\|C\|\|\operatorname{diag}((1-u)a_{\lambda_{j}})\|.$

Using (6), we thus find

(12) $\|(1-u)d\|\leq\kappa^{2}\varepsilon.$

On the other hand, using that $q$ commutes with $A$ we may write $qd=qdq$, and
by (8),(9) and (10) we find that

$\displaystyle|\tau((qd-q)xy^{*})|$
$\displaystyle=|\tau((qdq-q)xy^{*})|=|\tau((qR\Delta Cq-q)xy^{*})|$
$\displaystyle=|\tau((qS\Delta Tq-q)xy^{*})|<\varepsilon+|\tau((V\Delta
V^{*}-q)xy^{*})|$ $\displaystyle<(1+\|x\|)\varepsilon$

whence

(13) $|\tau((qd-q)xy^{*})|<(1+\kappa)\varepsilon.$

Note that $dx\in A$ so that $\varphi(dx)=0$ since $\varphi\in A^{\perp}$.
Therefore,

$\displaystyle\varphi(qx)$ $\displaystyle=\varphi((q-d)x)=\tau((q-d)xy^{*})$
$\displaystyle=\tau((q-qd)xy^{*})-\tau((u-q)dxy^{*})-\tau((1-u)dxy^{*})$

so by Lemma 2.1 we find

$|\varphi(qx)|\leq|\tau((qd-q)xy^{*})|+\|x\|\|d\|\tau(u-q)^{1/2}+\|x\|\|(1-u)d\|.$

Combined with (5),(11),(12) and (13), this in turn implies

$\displaystyle\tau(q)$
$\displaystyle<\varepsilon+|\varphi(qx)|\leq\varepsilon+(1+\kappa)\varepsilon+\kappa^{3}\tau(u-q)^{1/2}+\kappa^{3}\varepsilon.$

Letting $\varepsilon\to 0$ yields the desired conclusion. ∎

###  4.1. Atomic parts and double annihilators

Theorem 4.3 makes it advantageous to determine when the atomic part of a
projection lies in the double annihilator of a subalgebra. As far as we know,
there are no available general criteria guaranteeing this. We thus set out to
identify a class of examples where this property is satisfied.

Our class of examples will involve tensor products. We recall some relevant
facts here, since several preliminaries are needed. Let $B$ and $D$ be
$\mathrm{C}^{*}$-algebras such that $B$ is nuclear. It follows from [AB1980,
Theorem 3.1] that there is an injective $*$-homomorphism
$\sigma:B^{**}\otimes_{\min}D^{**}\to(B\otimes_{\min}D)^{**}$ with the
following properties.

  1. (i)

$\sigma(b\otimes d)=b\otimes d$ for each $b\in B$ and $d\in D$.

  2. (ii)

Let $\sigma_{B}:B^{**}\to(B\otimes_{\min}D)^{**}$ and
$\sigma_{D}:D^{**}\to(B\otimes_{\min}D)^{**}$ be the restriction
$*$-homomorphisms arising from $\sigma$ (see for instance [BO2008, Theorem
3.2.6]). Then, $\sigma_{B}$ and $\sigma_{D}$ are both weak-$*$ continuous.

Via this map $\sigma$, we will view $B^{**}\otimes_{\min}D^{**}$ as a
$\mathrm{C}^{*}$-subalgebra of $(B\otimes_{\min}D)^{**}$. In particular, given
projections $q\in B^{**}$ and $r\in D^{**}$, $q\otimes r$ is a projection in
$(B\otimes_{\min}D)^{**}$. For the following, recall that if $A\subset B$ is a
subalgebra, then

$A\otimes_{\min}D\subset B\otimes_{\min}D$

by injectivity of the minimal tensor product.

######  Lemma 4.4.

Let $B$ and $D$ be $\mathrm{C}^{*}$-algebras such that $B$ is nuclear. Let
$q\in B^{**}$ and $r\in D^{**}$ be projections, with $r$ open or closed. Let
$A\subset B$ be a closed subalgebra and assume that $q\in A^{\perp\perp}$.
Then, $q\otimes r\in(A\otimes D)^{\perp\perp}$.

###### Proof.

We first assume that $r$ is open. Let $(d_{j})$ be a net of positive
contractions in $D$ increasing to $r$ in the weak-$*$ topology of $D^{**}$.
Then $(\sigma_{D}(d_{j}))$ increases to $\sigma_{D}(r)$ in the weak-$*$
topology of $(B\otimes_{\min}D)^{**}$. On the other hand, by Goldstine’s
theorem there is a net $(a_{i})$ of contractions in $A$ converging to $q$ in
the weak-$*$ topology of $B^{**}$. We see that the net
$(\sigma_{B}(a_{i})\sigma_{D}(d_{j}))_{(i,j)}$ converges to $\sigma(q\otimes
r)$ in the weak-$*$ topology of $(B\otimes_{\min}D)^{**}$, being the product
of a strongly converging and a weak-$*$ converging net. Each
$\sigma_{B}(a_{i})\sigma_{D}(d_{j})=\sigma(a_{i}\otimes d_{j})$ lies in the
image of $A\otimes D$ inside of $(B\otimes_{\min}D)^{**}$, and therefore
$q\otimes r\in(A\otimes_{\min}D)^{\perp\perp}$.

If $r$ is closed, then by the above $q\otimes(1-r)$ lies in
$(A\otimes_{\min}D)^{\perp\perp}$, and hence so does $q\otimes r$. ∎

Before proving the final preliminary result, recall that for a projection
$p\in B^{**}$, the following statements are equivalent.

  1. (i)

The projection $p$ is minimal in $B^{**}$.

  2. (ii)

$pB^{**}p={\mathbb{C}}p$.

  3. (iii)

There is a pure state $\omega$ on $B$ such that $\omega(b)p=pbp$ for each
$b\in B$.

Using this equivalence, it is easily verified that if $p\in B^{**}$ and $q\in
D^{**}$ are minimal projections, then the projection $p\otimes q$ is also
minimal in $(B\otimes_{\min}D)^{**}$.

######  Lemma 4.5.

Let $B$ and $D$ be $\mathrm{C}^{*}$-algebras such that $B$ is commutative. Let
$p\in(B\otimes_{\min}D)^{**}$ be a minimal projection. Then, there are minimal
projections $e\in B^{**},f\in D^{**}$ such that $p=e\otimes f$.

###### Proof.

There is a pure state $\omega$ on $B\otimes_{\min}D$ with support projection
equal to $p$. By [takesaki2002, Theorem IV.4.14], there are pure states
$\varphi$ of $B$ and $\psi$ of $D$ such that $\omega=\varphi\otimes\psi$. Let
$e\in B^{**}$ and $f\in D^{**}$ denote the support projections of $\varphi$
and $\psi$, respectively. These are both minimal, so that $e\otimes f$ is
minimal as well. On the other hand, we have $\omega(e\otimes f)=1$, so
$e\otimes f\geq p$, which then forces $p=e\otimes f$. ∎

We now arrive at our class of examples where all projections have their atomic
parts in the double annihilator of a subalgebra.

######  Proposition 4.6.

Let $X$ be a compact metric space and let $A\subset\mathrm{C}(X)$ be a unital
norm-closed subalgebra separating the points of $X$. Assume that $X$ is the
Choquet boundary of $A$. Let $D$ be a unital $\mathrm{C}^{*}$-algebra. Then,
every minimal projection in $(\mathrm{C}(X)\otimes_{\min}D)^{**}$ lies in
$(A\otimes_{\min}D)^{\perp\perp}$. In particular, the atomic part of any
projection in $(\mathrm{C}(X)\otimes_{\min}D)^{**}$ lies in
$(A\otimes_{\min}D)^{\perp\perp}$.

###### Proof.

Let $q\in(\mathrm{C}(X)\otimes_{\min}D)^{**}$ be a minimal projection. By
Lemma 4.5, there are minimal projections $e\in\mathrm{C}(X)^{**}$ and $f\in
D^{**}$ such that $q=e\otimes f$. On the other hand, by [gamelin1969, Theorem
II.11.3], we see that every $x\in X$ is an $A$-peak point. In other words,
every minimal projection in $\mathrm{C}(X)^{**}$ lies in $A^{\perp\perp}$. In
particular, this means that $e\in A^{\perp\perp}$, so by Lemma 4.4, we
conclude that $p\in(A\otimes_{\min}D)^{\perp\perp}$. This proves the first
statement. The second statement immediately follows from the first along with
(2). ∎

###  4.2. Recovering Varopoulos’ null projections

We identify special cases where the conclusion of Theorem 4.3 can be
strengthened to recover the fact that $q$ is $A$-null, as in Varopoulos’
original result. More precisely, we impose a commutation condition between $A$
and the net approximating the closed projection $q$.

######  Corollary 4.7.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
norm-closed subalgebra. Let $q\in B^{**}$ be a projection with the following
properties:

  1. (i)

there is a net $(b_{i})$ of positive contractions in $B$ commuting with $A$
and decreasing to $q$ in the weak-$*$ topology of $B^{**}$,

  2. (ii)

there is $\kappa>0$ for which $q$ is a complete $(A,\kappa)$-interpolation
projection, and

  3. (iii)

the atomic part of $q$ lies in $A^{\perp\perp}$.

Then, $q$ is $A$-null.

###### Proof.

First note that assumption (i) implies that $q$ is closed and commutes with
$A$. Let $\varepsilon>0$ and $\varphi\in A^{\perp}$ with $\|\varphi\|=1$. Let
$C\subset B$ denote the $\mathrm{C}^{*}$-subalgebra generated by $1$ and the
elements $b_{i}$. Hence $A$ and $C$ commute. By virtue of [akemann1969,
Proposition II.3] (and its proof), there is an open projection $u\in B^{**}$
lying in $C^{\perp\perp}$ such that $|\varphi|(u-q)<\varepsilon$. In
particular, $u$ commutes with $A$, and hence is $A$-invariant. Theorem 4.3
then gives $|\varphi|(q)\leq\kappa^{3}\sqrt{\varepsilon}$, and the proof is
complete upon letting $\varepsilon\to 0$. ∎

Next, we show how the previous result implies Varopoulos’ original statement.
Before proceeding with the proof, we record a useful fact about interpolation
projections, inspired by the trick used in [DH2023, Lemma 10.1].

######  Proposition 4.8.

Let $B$ be a unital nuclear $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a
unital norm-closed subalgebra. Let $q\in B^{**}$ be a central closed
projection such that $Aq=Bq$ as subspaces in $B^{**}$. Then, there is
$\kappa>0$ such that $q$ is a complete $(A,\kappa)$-interpolation projection.

###### Proof.

Consider the surjective completely contractive map $\pi:A\to Bq$ defined as
$\pi(a)=aq$ for each $a\in A$. Since $q$ is central, $Bq$ is a
$\mathrm{C}^{*}$-algebra and $\pi$ is a homomorphism. Therefore, there is a
bounded homomorphism $\rho:Bq\to A/\ker\pi$ such that $\rho\circ\pi:A\to
A/\ker\pi$ is simply the quotient map. Next, since $B$ is nuclear, so is $Bq$
[BO2008, Corollary 9.4.4]. In addition, because $A/\ker\pi$ is an operator
algebra (see [BLM2004, Proposition 2.3.4]), we may apply [christensen1981,
Theorem 4.1] to see that $\rho$ is in fact completely bounded. This
immediately implies the existence of $\kappa>0$ satisfying the required
property. ∎

We note in passing that in view of [LT2012], one-sided module maps on nuclear
$\mathrm{C}^{*}$-algebras do not have to be completely bounded.

We can now show how to recover Varopoulos’ original theorem from our general
machinery.

######  Corollary 4.9.

Let $X$ be a compact Hausdorff space and let $A\subset\mathrm{C}(X)$ be a
unital norm-closed subalgebra. Let $K\subset X$ be a subset with the following
properties:

  1. (i)

$K$ is closed,

  2. (ii)

$A\chi_{K}=\mathrm{C}(X)\chi_{K}$ as subspaces in $\mathrm{C}(X)^{**}$ (that
is, $K$ is an interpolation set), and

  3. (iii)

$\chi_{\\{x\\}}\in A^{\perp\perp}$ for each $x\in K$ (that is, every point in
$K$ is an $A$-peak point).

Then, $\chi_{K}$ is $A$-null (that is, $K$ is a peak set).

###### Proof.

Let $q=\chi_{K}$, which is a closed projection in $\mathrm{C}(X)^{**}$.
Invoking (2), we see that the atomic part of $\chi_{K}$ is simply
$\bigvee_{x\in K}\chi_{\\{x\\}}$. Given $x\in K$, we know that
$\chi_{\\{x\\}}\in A^{\perp\perp}$. Hence, the atomic part of $\chi_{K}$ lies
in $A^{\perp\perp}$. Next, it follows from Proposition 4.8 that there is
$\kappa>0$ such that $q$ is a complete $(A,\kappa)$-interpolation projection.
By virtue of Corollary 4.7, we conclude that $q$ is $A$-null. ∎

##  5\. Application to algebras of analytic functions

In this section, we illustrate how our general theorem can be used to recover
some recent results pertaining to operator algebras of analytic functions on
the unit ball [DH2023]. This will be done through another special case of
Theorem 4.3, which we tackle first.

###  5.1. Adapted projections

Let $B$ be a $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a subalgebra. We
say that a projection $e\in B^{**}$ is _$A$ -adapted_ if there is a net
$(e_{i})$ of closed $A$-coinvariant projections in $B^{**}$ increasing to $e$
in the weak-$*$ topology of $B^{**}$. The linchpin technical result of this
section is the following refinement of Theorem 4.3.

######  Corollary 5.1.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
norm-closed subalgebra. Assume that there is a central closed projection $z\in
B^{**}$ such that $B^{**}z$ is commutative and $1-z$ is $A$-adapted. Let $q\in
B^{**}$ be a projection with the following properties:

  1. (i)

$q$ is closed,

  2. (ii)

there is $\kappa>0$ for which $q$ is a complete $(A,\kappa)$-interpolation
projection,

  3. (iii)

$q\leq z$,

  4. (iv)

the atomic part of $q$ lies in $A^{\perp\perp}$.

Then, $q$ is $A$-null.

###### Proof.

Let $\varepsilon>0$ and let $\varphi\in A^{\perp}$ with $\|\varphi\|=1$. Let
$\tau=|\varphi|$, which is a state on $B$. By [akemann1969, Proposition II.3],
there is an open projection $u\in B^{**}$ dominating $q$ such that
$\tau(u-q)<\varepsilon.$ Since $z$ is central and $q\leq z$, we find $q=qz\leq
uz\leq u$ so that

(14) $\tau(uz-q)<\varepsilon.$

Further, because $1-z$ is $A$-adapted, there is an $A$-coinvariant closed
projection $r\leq 1-z$ such that

(15) $\tau((1-z)-r)<\varepsilon.$

Because $z$ is closed, so is $z(1-u)$ [akemann1969, Proposition II.5].
Moreover, we have $r\leq 1-z$ so that $r^{\prime}=r+z(1-u)$ is closed by
[akemann1969, Theorem II.7]. The projection

$u^{\prime}=1-r^{\prime}=((1-z)-r))+uz$

is then seen to be open with $u^{\prime}\geq uz\geq q$. We compute

$\displaystyle\tau(u^{\prime}-q)$ $\displaystyle=\tau(uz-q)+\tau((1-z)-r)\leq
2\varepsilon$

by (14) and (15).

Observe next that since $B^{**}z$ is commutative, $q=qz$ is $A$-coinvariant
and $uz$ is $A$-invariant. On the other hand, $1-z$ is central and $r$ is
$A$-coinvariant, so that $(1-z)-r$ is $A$-invariant. It follows that
$u^{\prime}$ is $A$-invariant as well. By virtue of Theorem 4.3, we see that

$\tau(q)\leq\kappa^{3}\sqrt{\tau(u^{\prime}-q)}\leq\kappa^{3}\sqrt{2\varepsilon},$

so the proof is complete upon letting $\varepsilon\to 0$. ∎

###  5.2. Hilbert function spaces and their multipliers

Our goal is to show how Corollary 5.1 can be used to recover [DH2023, Theorem
10.3]. Along the way, we will also recover a few more results from [DH2023].
Let us recall some background material and terminology; for more detail the
reader may consult [DH2023] or e.g. [CT2023Henkin, Section 5].

Let $d\geq 1$ be a positive integer and let
${\mathbb{B}}_{d}\subset{\mathbb{C}}^{d}$ denote the open unit ball. Let
${\mathbb{S}}_{d}=\partial{\mathbb{B}}_{d}$ denote the unit sphere. Let
${\mathcal{H}}$ be a Hilbert space of analytic functions on ${\mathbb{B}}_{d}$
admitting a reproducing kernel that is unitarily invariant and regular.
Examples include the Dirichlet space on the disc, the Hardy and Bergman
spaces, as well as the Drury–Arveson space. For each
$\lambda\in{\mathbb{B}}_{d}$, we let $k_{\lambda}\in{\mathcal{H}}$ denote the
corresponding reproducing kernel vector. We then have that
$\\{k_{\lambda}:\lambda\in{\mathbb{B}}_{d}\\}$ spans a norm-dense subspace in
${\mathcal{H}}$ (see [GHX04]).

A multiplier is a function $\varphi:{\mathbb{B}}_{d}\to{\mathbb{C}}$
satisfying $\varphi f\in{\mathcal{H}}$ for every $f\in{\mathcal{H}}$. The
corresponding multiplication operator
$M_{\varphi}:{\mathcal{H}}\to{\mathcal{H}}$ is then bounded. The space of all
multiplication operators induced by a multiplier is denoted by
${\mathcal{M}}({\mathcal{H}})$, and it is weak-$*$ closed subalgebra of
$B({\mathcal{H}})$. If $\varphi$ is a multiplier, then

(16)
$M_{\varphi}^{*}k_{\lambda}=\overline{\varphi(\lambda)}k_{\lambda},\quad\lambda\in{\mathbb{B}}_{d}.$

All polynomials are multipliers, and we let
${\mathcal{A}}({\mathcal{H}})\subset{\mathcal{M}}({\mathcal{H}})$ denote the
norm-closure of these polynomial multipliers. A standard argument shows that
if $M_{\varphi}\in{\mathcal{A}}({\mathcal{H}})$, then $\varphi$ is continuous
on $\overline{{\mathbb{B}}_{d}}$. Indeed, for any multiplier $\varphi$, (16)
shows that $\|\varphi\|_{\infty}\leq\|M_{\varphi}\|$; hence functions inducing
the multiplication operators in ${\mathcal{A}}({\mathcal{H}})$ are uniform
limits of polynomials on $\overline{{\mathbb{B}}_{d}}$.

Denote by ${\mathcal{T}}({\mathcal{H}})$ the $\mathrm{C}^{*}$-subalgebra of
$B({\mathcal{H}})$ generated by${\mathcal{A}}({\mathcal{H}})$. It follows from
[GHX04, Theorem 4.6] that ${\mathcal{T}}({\mathcal{H}})$ contains the ideal
$K$ of compact operators on ${\mathcal{H}}$. Let
$z_{K}\in{\mathcal{T}}({\mathcal{H}})^{**}$ denote the closed central
projection satisfying
${\mathcal{T}}({\mathcal{H}})^{**}(1-z_{K})=K^{\perp\perp}$. This projection
has the following crucial property.

######  Lemma 5.2.

The projection $1-z_{K}$ is ${\mathcal{A}}({\mathcal{H}})$-adapted.

###### Proof.

For each finite subset $F\subset{\mathbb{B}}_{d}$, let $e_{F}\in
B({\mathcal{H}})$ denote the projection onto the subspace spanned by
$\\{k_{\lambda}:\lambda\in F\\}$. Since the set
$\\{k_{\lambda}:\lambda\in{\mathbb{B}}_{d}\\}$ spans a norm-dense subset of
${\mathcal{H}}$, the net $(e_{F})$ increases to the identity on
${\mathcal{H}}$ in the weak-$*$ topology of $B({\mathcal{H}})$. Further, each
$e_{F}$ is compact on ${\mathcal{H}}$, and it is
${\mathcal{A}}({\mathcal{H}})$-coinvariant by virtue of (16). Next, by
[CT2023Henkin, Lemma 5.5], there is a weak-$*$ homeomorphic $*$-isomorphism
$\Omega:{\mathcal{T}}({\mathcal{H}})^{**}(1-z_{K})\to B({\mathcal{H}})$ such
that $\Omega(t(1-z_{K}))=t$ for every $t\in{\mathcal{T}}({\mathcal{H}})$.
Since $e_{F}$ is compact, it lies in ${\mathcal{T}}({\mathcal{H}})$ and we see
that $\Omega^{-1}(e_{F})=e_{F}(1-z_{K})=e_{F}$. It follows that
$(\Omega^{-1}(e_{F}))$ is an increasing net of
${\mathcal{A}}({\mathcal{H}})$-coinvariant closed projections in
${\mathcal{T}}({\mathcal{H}})^{**}$ increasing to $1-z_{K}$ in the weak-$*$
topology of ${\mathcal{T}}({\mathcal{H}})^{**}$. ∎

Let $\gamma:{\mathcal{T}}({\mathcal{H}})\to{\mathcal{T}}({\mathcal{H}})/K$
denote the quotient map. There is a natural $*$-isomorphism
$\tau:{\mathcal{T}}({\mathcal{H}})/K\to\mathrm{C}({\mathbb{S}}_{d})$ such that

$(\tau\circ\gamma)(M_{\varphi})=\varphi,\quad
M_{\varphi}\in{\mathcal{A}}({\mathcal{H}}).$

See e.g. [GHX04]. In particular, ${\mathcal{T}}({\mathcal{H}})$ is nuclear
[BO2008, Exercise 3.8.1]. The restriction of $(\tau\circ\gamma)^{**}$ gives a
weak-$*$ homeomorphic $*$-isomorphism
$\Theta:{\mathcal{T}}({\mathcal{H}})^{**}z_{K}\to\mathrm{C}({\mathbb{S}}_{d})^{**}$.
We now give a description of the closed projections in
${\mathcal{T}}({\mathcal{H}})^{**}$ dominated by $z_{K}$.

######  Lemma 5.3.

Let $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ be a projection. Then, the
following statements hold.

  1. (i)

If $q$ is closed, then there is a closed subset $E\subset{\mathbb{S}}_{d}$
such that $(\tau\circ\gamma)^{**}(q)=\chi_{E}$.

  2. (ii)

Assume that $q\leq z_{K}$. If there is a closed subset
$E\subset{\mathbb{S}}_{d}$ such that $\Theta(q)=\chi_{E}$, then $q$ is closed.

###### Proof.

(i) Let $(t_{i})$ be a net of positive contractions in
${\mathcal{T}}({\mathcal{H}})$ decreasing to $q$ in the weak-$*$ topology of
${\mathcal{T}}({\mathcal{H}})^{**}$. Since $\Theta$ is a weak-$*$ continuous
$*$-homomorphism with $\Theta(z_{K})=1$, it follows that
$(\Theta(t_{i}z_{K}))$ is a net of positive contractions in
$\mathrm{C}({\mathbb{S}}_{d})$ decreasing to $\Theta(q)$ in the weak-$*$
topology of $\mathrm{C}({\mathbb{S}}_{d})^{**}$. In other words, $\Theta(q)$
is closed in $\mathrm{C}({\mathbb{S}}_{d})^{**}$, so there is a closed subset
$E\subset{\mathbb{S}}_{d}$ such that $\Theta(q)=\chi_{E}$.

(ii) Since $1-\chi_{E}$ is open in $\mathrm{C}({\mathbb{S}}_{d})^{**}$, there
is a closed two-sided ideal $J\subset\mathrm{C}({\mathbb{S}}_{d})$ such that
$J^{\perp\perp}=\mathrm{C}({\mathbb{S}}_{d})^{**}(1-\chi_{E})$. Let
$\pi:\mathrm{C}({\mathbb{S}}_{d})\to\mathrm{C}({\mathbb{S}}_{d})/J$ denote the
corresponding quotient map. Then,

$\displaystyle\ker(\pi\circ\tau\circ\gamma)^{\perp\perp}$
$\displaystyle=\ker(\pi\circ\tau\circ\gamma)^{**}={\mathcal{T}}({\mathcal{H}})^{**}(1-z_{K})+{\mathcal{T}}({\mathcal{H}})^{**}(z_{K}-q)$
$\displaystyle={\mathcal{T}}({\mathcal{H}})^{**}(1-q).$

We conclude that $q$ is closed in ${\mathcal{T}}({\mathcal{H}})^{**}$. ∎

A closed set $E\subset{\mathbb{S}}_{d}$ is an _${\mathcal{A}}({\mathcal{H}})$
-interpolation set_ if

$\\{\varphi:M_{\varphi}\in{\mathcal{A}}({\mathcal{H}})\\}\,\chi_{E}=\mathrm{C}({\mathbb{S}}_{d})\,\chi_{E}$

as subsets of $\mathrm{C}({\mathbb{S}}_{d})^{**}$. This is related to our
notion of interpolation projection in ${\mathcal{T}}({\mathcal{H}})^{**}$, as
follows.

######  Lemma 5.4.

Let $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ be a closed projection dominated
by $z_{K}$. Then, the following statements are equivalent.

  1. (i)

${\mathcal{A}}({\mathcal{H}})q={\mathcal{T}}({\mathcal{H}})q$.

  2. (ii)

There is a closed ${\mathcal{A}}({\mathcal{H}})$-interpolation subset
$E\subset{\mathbb{S}}_{d}$ such that $\Theta(q)=\chi_{E}$.

###### Proof.

By Lemma 5.3, there is a closed subset $E\subset{\mathbb{S}}_{d}$ such that
$\Theta(q)=\chi_{E}$. Since $\Theta$ is a $*$-isomorphism such that

$\Theta(M_{\varphi}z_{K})=\varphi,\quad
M_{\varphi}\in{\mathcal{A}}({\mathcal{H}}),$

we infer that ${\mathcal{A}}({\mathcal{H}})q={\mathcal{T}}({\mathcal{H}})q$ is
equivalent to
$\Theta({\mathcal{A}}({\mathcal{H}})q)=\Theta({\mathcal{T}}({\mathcal{H}})q)$,
or

$\\{\varphi:M_{\varphi}\in{\mathcal{A}}({\mathcal{H}})\\}\chi_{E}=\mathrm{C}({\mathbb{S}}_{d})\chi_{E}.\qed$

A bounded linear functional
$\varphi:{\mathcal{A}}({\mathcal{H}})\to{\mathbb{C}}$ will be said to be
_${\mathcal{M}}({\mathcal{H}})$ -Henkin_ if $\varphi$ extends weak-$*$
continuously to ${\mathcal{M}}({\mathcal{H}})$. We note here that this
coincides with the notion used in [CT2023Henkin] by Equation (6) above Lemma
5.6 there (or [DH2023, Lemma 2.1]). This will be used implicitly below,
whenever results from [CT2023Henkin] regarding Henkin functionals are invoked.

A regular Borel measure $\mu$ on ${\mathbb{S}}_{d}$ is said to be
${\mathcal{M}}({\mathcal{H}})$-Henkin if
$I_{\mu}\circ\tau\circ\gamma|_{{\mathcal{A}}({\mathcal{H}})}$ is
${\mathcal{M}}({\mathcal{H}})$–Henkin in the previous sense, where
$I_{\mu}:\mathrm{C}({\mathbb{S}}_{d})\to{\mathbb{C}}$ is the integration
functional defined as

$I_{\mu}(f)=\int_{{\mathbb{S}}_{d}}fd\mu,\quad
f\in\mathrm{C}({\mathbb{S}}_{d}).$

A closed projection $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ is
_${\mathcal{M}}({\mathcal{H}})$ -totally null_ if $|\varphi|(q)=0$ for every
functional $\varphi\in{\mathcal{T}}({\mathcal{H}})^{*}$ such that
$\varphi|_{{\mathcal{A}}({\mathcal{H}})}$ is
${\mathcal{M}}({\mathcal{H}})$-Henkin. A Borel subset
$E\subset{\mathbb{S}}_{d}$ is said to be
${\mathcal{M}}({\mathcal{H}})$-totally null if $|\mu|(E)=0$ for every
${\mathcal{M}}(H)$-Henkin measure. We clarify the relation between these
notions in the next result. In what follows, we will repeatedly use the fact
that $\varphi(\cdot(1-z_{K}))|_{{\mathcal{A}}({\mathcal{H}})}$ is an
${\mathcal{M}}({\mathcal{H}})$-Henkin functional for every
$\varphi\in{\mathcal{T}}({\mathcal{H}})^{*}$ (see [CT2023Henkin, Lemma 5.5]).

######  Theorem 5.5.

Let ${\mathcal{H}}$ be a Hilbert space of analytic functions on
${\mathbb{B}}_{d}$ admitting a reproducing kernel that is unitarily invariant
and regular. Let $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ be a projection.
Then, the following statements are equivalent.

  1. (i)

The projection $q$ is closed and ${\mathcal{A}}({\mathcal{H}})$-null.

  2. (ii)

The projection $q$ is closed and ${\mathcal{M}}({\mathcal{H}})$-totally null.

  3. (iii)

The projection $q$ is dominated by $z_{K}$, and there is a closed
${\mathcal{M}}({\mathcal{H}})$-totally null subset $E\subset{\mathbb{S}}_{d}$
such that $\Theta(q)=\chi_{E}$.

###### Proof.

(i) $\Leftrightarrow$ (ii): By Lemma 5.6 in [CT2023Henkin] and Equation (6)
above it, we see that [CT2023Henkin, Theorem 6.2] may be applied to the triple
${\mathcal{A}}({\mathcal{H}})\subset{\mathcal{T}}({\mathcal{H}})\subset
B({\mathcal{H}})$, which in turn gives the desired equivalence.

(ii) $\Rightarrow$ (iii): Let $\varphi\in{\mathcal{T}}({\mathcal{H}})$ be a
state. Put $\psi=\varphi(\cdot(1-z_{K}))$, which is positive since $z_{K}$ is
central. Using that $\psi|_{{\mathcal{A}}({\mathcal{H}})}$ is an
${\mathcal{M}}({\mathcal{H}})$-Henkin functional, we find $\psi(q)=0$, or
$\varphi(q(1-z_{K}))=0$. Since $\varphi$ was arbitrary, we infer that
$q(1-z_{K})=0$ or $q\leq z_{K}$.

By Lemma 5.3, there is a closed set $E\subset{\mathbb{S}}_{d}$ such that
$\chi_{E}=\Theta(q)$. We claim that $E$ is
${\mathcal{M}}({\mathcal{H}})$-totally null. To see this, let $\mu$ be an
${\mathcal{M}}({\mathcal{H}})$-Henkin measure. Then,
$|I_{\mu}\circ\tau\circ\gamma|(q)=0$. By [CT2023Henkin, Lemma 2.3] we find
$|I_{\mu}\circ\tau\circ\gamma|=|I_{\mu}|\circ\tau^{**}\circ\gamma^{**}$ so

$\displaystyle|\mu|(E)$
$\displaystyle=|I_{\mu}|(\chi_{E})=(|I_{\mu}|\circ\Theta)(q)=(|I_{\mu}|\circ\tau^{**}\circ\gamma^{**})(q)=|I_{\mu}\circ\tau\circ\gamma|(q)=0$

which establishes the claim.

(iii) $\Rightarrow$ (ii): The fact that $q$ is closed follows from Lemma 5.3.
Let $\varphi\in{\mathcal{T}}({\mathcal{H}})^{*}$ be such that
$\varphi|_{{\mathcal{A}}({\mathcal{H}})}$ is an
${\mathcal{M}}({\mathcal{H}})$-Henkin functional. The goal is to show that
$|\varphi|(q)=0$. By assumption, we have $q\leq z_{K}$, so it is equivalent to
show that $|\varphi|(qz_{K})=0$. Using [CT2023Henkin, Theorem 5.7], it follows
that the restriction to ${\mathcal{A}}({\mathcal{H}})$ of
$\psi=|\varphi|(\cdot z_{K})$ is also an ${\mathcal{M}}({\mathcal{H}})$-Henkin
functional. The functional $\psi\circ\Theta^{-1}$ is bounded and positive on
$\mathrm{C}({\mathbb{S}}_{d})$, so there is a regular positive Borel measure
$\mu$ on ${\mathbb{S}}_{d}$ such that
$I_{\mu}=(\psi\circ\Theta^{-1})|_{\mathrm{C}({\mathbb{S}}_{d})}$. By
construction, $\mu$ is an ${\mathcal{M}}({\mathcal{H}})$-Henkin measure, so
that $\mu(E)=0$.

Since $q$ is closed and ${\mathcal{T}}({\mathcal{H}})$ is finitely generated
by the monomials, hence is separable, we may choose a sequence of positive
contractions $(t_{n})$ in ${\mathcal{T}}({\mathcal{H}})$ decreasing to $q$ in
the weak-$*$ topology of ${\mathcal{T}}({\mathcal{H}})^{**}$. In particular,
we see that the bounded sequence of continuous functions
$((\tau\circ\gamma)(t_{n}))$ converges pointwise on ${\mathbb{S}}_{d}$ to
$\chi_{E}$. By dominated convergence and using that
$(\tau\circ\nu)^{**}(z_{K})=1$, we then find

$\displaystyle\psi(q)$
$\displaystyle=\lim_{n\to\infty}\psi(t_{n}z_{K})=\lim_{n\to\infty}(\psi\circ\Theta^{-1})((\tau\circ\gamma)(t_{n})))$
$\displaystyle=\lim_{n\to\infty}\int_{{\mathbb{S}}_{d}}(\tau\circ\gamma)(t_{n})d\mu=\int_{{\mathbb{S}}_{d}}\chi_{E}d\mu=\mu(E)=0.$

We conclude that $q$ is indeed ${\mathcal{M}}({\mathcal{H}})$-totally null. ∎

###  5.3. The Davidson–Hartz theorems

The previous result has bearing on the topic of peak-interpolation. Indeed, it
allows us to provide alternative proofs (and even some refinements) of some of
the results from [DH2023] via the noncommutative peak-interpolation theory
surveyed in [blecher2013]. For the most part, the statement below is a
commutative one. That our approach takes us through the abstract
noncommutative theory is a novel feature that we feel has potential for
further future applications. We thank Ken Davidson and Michael Hartz for
several enjoyable and helpful discussions on this topic.

######  Theorem 5.6.

Let ${\mathcal{H}}$ be a Hilbert space of analytic functions on
${\mathbb{B}}_{d}$ admitting a reproducing kernel that is unitarily invariant
and regular. Let $E\subset{\mathbb{S}}_{d}$ be a closed
${\mathcal{M}}({\mathcal{H}})$-totally null set. Then, for every non-zero
function $g\in\mathrm{C}(E)$ and every strictly positive function
$k\in\mathrm{C}({\mathbb{S}}_{d})$ such that $|g|\leq k$ on $E$, there is
$M_{\varphi}\in{\mathcal{A}}({\mathcal{H}})$ with the following properties:

  1. (i)

$\|M_{\varphi}\|=\|g\|$,

  2. (ii)

$\varphi|_{E}=g$,

  3. (iii)

$|\varphi|\leq k$ on ${\mathbb{S}}_{d}$, and

  4. (iv)

$|\varphi|<\|g\|$ on ${\mathbb{B}}_{d}$.

###### Proof.

Let $g\in\mathrm{C}(E)$ with $\|g\|=1$. Let
$k_{0}\in\mathrm{C}({\mathbb{S}}_{d})$ be a strictly positive function such
that $k_{0}|_{E}=1$ and $0<k<1$ on ${\mathbb{S}}_{d}\setminus E$. Let
$k\in\mathrm{C}({\mathbb{S}}_{d})$ be strictly positive such that $|g|\leq k$
on $E$ and put $k_{1}=\min\\{k,k_{0}\\}$. By Tietze’s extension theorem, there
is $g_{0}\in\mathrm{C}({\mathbb{S}}_{d})$ such that $g_{0}|_{E}=g$ and
$\|g_{0}\|=1$. Recall that
$\tau\circ\gamma:{\mathcal{T}}({\mathcal{H}})\to\mathrm{C}({\mathbb{S}}_{d})$
is a surjective $*$-homomorphism. Thus, there is
$b\in{\mathcal{T}}({\mathcal{H}})$ with $\|b\|=1$ and a positive element
$d\in{\mathcal{T}}({\mathcal{H}})$ with $\|d\|=\|k_{1}\|$ such that
$(\tau\circ\gamma)(b)=g_{0}$ and and $(\tau\circ\gamma)(d)=k_{1}$.

Next, by Lemma 5.3 there is a closed projection $q\leq z_{K}$ such that
$\Theta(q)=\chi_{E}$. Note that $q$ is central in
${\mathcal{T}}({\mathcal{H}})^{**}$ since
${\mathcal{T}}({\mathcal{H}})^{**}z_{K}$ is abelian. Moreover, $q$ is
${\mathcal{A}}({\mathcal{H}})$-null by Theorem 5.5. By construction, we have

$\displaystyle\Theta(d^{2}q)=k_{1}^{2}\chi_{E}\geq|g_{0}|^{2}\chi_{E}=\Theta(qb^{*}bq)$

so that $qb^{*}bq\leq d^{2}q$. Invoking now [blecher2013, Theorem 3.4] or
[blecher2022, Theorem 2.3 (iii)], we obtain $a\in{\mathcal{A}}({\mathcal{H}})$
with $aq=qa=bq$ and $a^{*}a\leq d^{2}$. There is a multiplier $\varphi$ such
that $a=M_{\varphi}$. We thus obtain

$\varphi\chi_{E}=\Theta(aq)=\Theta(bq)=g_{0}\chi_{E}$

whence $\varphi|_{E}=g$ and (ii) holds. Since $M_{\varphi}^{*}M_{\varphi}\leq
d^{2}$ we find $\|M_{\varphi}\|\leq\|d\|=\|k_{1}\|\leq 1=\|g\|$. This in turn
this forces $\|M_{\varphi}\|=\|g\|$, since

$\|M_{\varphi}\|\geq\|aq\|=\|bq\|=\|g_{0}\chi_{E}\|=\|g\|.$

Thus we have (i). In addition, applying $\Theta$ to the inequality
$M_{\varphi}^{*}M_{\varphi}\leq d^{2}$ implies $|\varphi|^{2}\leq|k_{1}|^{2}$
on ${\mathbb{S}}_{d}$, and hence (iii) is satisfied. Finally, because
$|\varphi|\leq k_{1}\leq k_{0}$, we infer that $\varphi$ is not constant on
${\mathbb{S}}_{d}$, and hence $|\varphi|<1$ on ${\mathbb{B}}_{d}$ by the
maximum modulus principle. ∎

The previous result is a sharpening of the scalar version of [DH2023, Theorem
8.1]. Indeed, it shows that peak-interpolation can be implemented while
maintaining domination by a control function, in line with Bishop’s original
theorem [bishop1962]. For this reason, let us say that a closed subset
$E\subset{\mathbb{S}}_{d}$ is a _Bishop peak-interpolation set_ if the
conclusion of Theorem 5.6 holds, that is every non-zero function $g\in C(E)$
may be ‘interpolated’ as in (i)–(iv) of the theorem. By Theorem 5.6 this class
of sets coincides with what are called peak interpolation sets in [DH2023].
One direction of this follows from (i)–(iv) with $k=\|g\|\,k_{0}$, where
$k_{0}$ is defined in the first line of the proof above. Conversely, any peak
set in the sense of [DH2023] satisfies Theorem 5.6 since it is
${\mathcal{M}}({\mathcal{H}})$-totally null by the standard argument in the
proof of [DH2023, Theorem 12.2]. So we see that all these classes, and the
peak sets, and the ${\mathcal{M}}({\mathcal{H}})$-totally null sets, coincide.

Next, we recover [DH2023, Theorem 10.3].

######  Theorem 5.7.

Let ${\mathcal{H}}$ be a Hilbert space of analytic functions on
${\mathbb{B}}_{d}$ admitting a reproducing kernel that is unitarily invariant
and regular. Let $E\subset{\mathbb{S}}_{d}$ be a closed
${\mathcal{A}}({\mathcal{H}})$-interpolation set. Assume that there exists a
non-empty ${\mathcal{M}}({\mathcal{H}})$-totally null set. Then, $E$ is
${\mathcal{M}}({\mathcal{H}})$-totally null.

###### Proof.

Consider the projection
$q=\Theta^{-1}(\chi_{E})\in{\mathcal{T}}({\mathcal{H}})^{**}z_{K}$, which is
closed by Lemma 5.3. To show that $E$ is
${\mathcal{M}}({\mathcal{H}})$-totally null, it is equivalent to show that $q$
is ${\mathcal{A}}({\mathcal{H}})$-null, by Theorem 5.5. Note that $1-z_{K}$ is
${\mathcal{A}}({\mathcal{H}})$-adapted by Lemma 5.2, and $q\leq z_{K}$. Thus,
the fact that $q$ is ${\mathcal{A}}({\mathcal{H}})$-null will follow from
Corollary 5.1, once we verify all the required conditions.

Using that $E$ is an ${\mathcal{A}}({\mathcal{H}})$-interpolation set, we find
${\mathcal{A}}({\mathcal{H}})q={\mathcal{T}}({\mathcal{H}})q$ by Lemma 5.4.
Since ${\mathcal{T}}({\mathcal{H}})$ is nuclear, by Proposition 4.8 we infer
that there is $\kappa>0$ such that $q$ is a complete
$({\mathcal{A}}({\mathcal{H}}),\kappa)$-interpolation projection.

It only remains to show that the atomic part of $q$ lies in
${\mathcal{A}}({\mathcal{H}})^{\perp\perp}$. Invoking (2), we see that the
atomic part of $\Theta(q)$ is simply $\bigvee_{\zeta\in E}\chi_{\\{\zeta\\}}$,
so that the atomic part of $q$ is $\bigvee_{\zeta\in E}p_{\\{\zeta\\}}$, with
$p_{\zeta}=\Theta^{-1}(\chi_{\\{\zeta\\}})$. Now, [DH2023, Lemma 9.1] implies
that the set $\\{\zeta\\}$ is ${\mathcal{M}}({\mathcal{H}})$-totally null set,
whence $p_{\zeta}$ is ${\mathcal{A}}({\mathcal{H}})$-null by Theorem 5.5, and
in particular $p_{\zeta}\in{\mathcal{A}}({\mathcal{H}})^{\perp\perp}$. ∎

The following result closes the circle of results about peak and null sets and
projections for ${\mathcal{A}}({\mathcal{H}})$. Recall that by [CT2023Henkin,
Lemma 5.5], there is a weak-$*$ homeomorphic $*$-isomorphism
$\Omega:{\mathcal{T}}({\mathcal{H}})^{**}(1-z_{K})\to B({\mathcal{H}})$ such
that $\Omega(t(1-z_{K}))=t$ for every $t\in{\mathcal{T}}({\mathcal{H}})$

######  Corollary 5.8.

Let ${\mathcal{H}}$ be a Hilbert space of analytic functions on
${\mathbb{B}}_{d}$ admitting a reproducing kernel that is unitarily invariant
and regular. Let $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ be a closed
projection. Then, the following statements are equivalent.

  1. (i)

$q$ is an ${\mathcal{A}}({\mathcal{H}})$-peak projection.

  2. (ii)

$q$ is ${\mathcal{A}}({\mathcal{H}})$-null.

  3. (iii)

$q$ is dominated by $z_{K}$ and there is a closed
${\mathcal{M}}({\mathcal{H}})$-totally null subset $E\subset{\mathbb{S}}_{d}$
such that $\Theta(q)=\chi_{E}$.

  4. (iv)

$q$ is dominated by $z_{K}$ and there is a closed Bishop peak-interpolation
subset $E\subset{\mathbb{S}}_{d}$ such that $\Theta(q)=\chi_{E}$.

If there exist non-empty ${\mathcal{M}}({\mathcal{H}})$-totally null sets,
then these statements are further equivalent to the following.

  1. (v)

$q$ is dominated by $z_{K}$ and there is a closed
${\mathcal{A}}({\mathcal{H}})$-interpolation set $E\subset{\mathbb{S}}_{d}$
such that $\Theta(q)=\chi_{E}$.

###### Proof.

(ii) $\Leftrightarrow$ (iii): This follows from Theorem 5.5.

(ii) $\Rightarrow$ (i): This follows from the “Basic principle” in the
introduction.

(i) $\Rightarrow$ (iii): By assumption, there is a contraction
$a\in{\mathcal{A}}({\mathcal{H}})$ such that $aq=q$ and $\|ap\|<1$ for every
closed projection $p\in{\mathcal{T}}({\mathcal{H}})^{**}$ with $pq=0$.
Consider the projection $P=\Omega(q(1-z_{K}))$ in $B({\mathcal{H}})$. Since
the sequence $(a^{n})$ converges to $q$ in the weak-$*$ topology of
${\mathcal{T}}({\mathcal{H}})^{**}$ [hay2007, Lemma 3.6], it follows that the
sequence $\Omega(a^{n})=a^{n}$ converges to $P$ in the weak-$*$ topology of
$B({\mathcal{H}})$. Then, any vector $f\in{\mathcal{H}}$ in the range of $P$
satisfies $af=f$. Since ${\mathcal{H}}$ consists of analytic functions on the
ball, by the identity theorem we conclude that $a=1$ if $P\neq 0$. But $a\neq
1$ since $q$ is ${\mathcal{A}}({\mathcal{H}})$-null. Therefore, $P=0$, and we
conclude that $(a^{n})$ converges to $0$ in the weak-$*$ topology of
$B({\mathcal{H}})$.

Next, let $\varphi\in{\mathcal{T}}({\mathcal{H}})^{*}$ be an
${\mathcal{M}}({\mathcal{H}})$-Henkin functional. By virtue of [CT2023Henkin,
Theorem 5.7], $|\varphi|$ is also ${\mathcal{M}}({\mathcal{H}})$-Henkin. By
definition, this implies that

$|\varphi|(q)=\lim_{n\to\infty}|\varphi|(a^{n})=0$

so $q$ is ${\mathcal{M}}({\mathcal{H}})$-totally null. We see that (iii)
follows from Theorem 5.5.

(iii) $\Rightarrow$ (iv): This is Theorem 5.6.

(iv) $\Rightarrow$ (iii): See the remark after Theorem 5.6.

It is trivial that (iv) $\Rightarrow$ (v), while (v)$\Rightarrow$ (iii)
follows from Theorem 5.7 under the assumption that there exist non-empty
${\mathcal{M}}({\mathcal{H}})$-totally null sets. ∎

We record some consequences of our arguments so far, using also the facts
established in the remark after Theorem 5.6.

######  Corollary 5.9.

Let ${\mathcal{H}}$ be a Hilbert space of analytic functions on
${\mathbb{B}}_{d}$ admitting a reproducing kernel that is unitarily invariant
and regular. The $*$-isomorphism
$\Theta:{\mathcal{T}}({\mathcal{H}})^{**}z_{K}\to\mathrm{C}({\mathbb{S}}_{d})^{**}$
restricts to an order preserving bijection between the class of closed
projections $q$ characterized in Corollary 5.8 and the class of Bishop peak-
interpolation (or equivalently, peak, or
${\mathcal{M}}({\mathcal{H}})$-totally null) subsets $E$ of
${\mathbb{S}}_{d}$.

Using this bijection one may add ‘interpolation set’ to the equivalences in
the last line, if nonempty peak sets exist. To close this section, we relate
the statements above to the condition that
${\mathcal{A}}({\mathcal{H}})q={\mathcal{T}}({\mathcal{H}})q$. The key
observation is the following. We say that a vector $\xi\in{\mathcal{H}}$ is
_strictly cyclic_ for ${\mathcal{A}}({\mathcal{H}})$ if
${\mathcal{A}}({\mathcal{H}})\xi={\mathcal{H}}$.

######  Lemma 5.10.

Let $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ be a closed projection such that
${\mathcal{A}}({\mathcal{H}})q={\mathcal{T}}({\mathcal{H}})q$. If
${\mathcal{A}}({\mathcal{H}})$ admits no strictly cyclic vector in
${\mathcal{H}}$, then $q\leq z_{K}$.

###### Proof.

If we put $P=\Omega(q(1-z_{K}))$ as before, then it suffices to show that
$P=0$. As subspaces in ${\mathcal{T}}({\mathcal{H}})^{**}$, we have
${\mathcal{A}}({\mathcal{H}})q={\mathcal{T}}({\mathcal{H}})q$, which in turn
implies
${\mathcal{A}}({\mathcal{H}})q(1-z_{K})={\mathcal{T}}({\mathcal{H}})q(1-z_{K})$.
Upon applying $\Omega$, we find
${\mathcal{A}}({\mathcal{H}})P={\mathcal{T}}({\mathcal{H}})P$ as subsets in
$B({\mathcal{H}})$. If $P\neq 0$, then there is a unit vector
$\xi\in{\mathcal{H}}$ in the range of $P$; let $E\in B({\mathcal{H}})$ denote
the corresponding rank-one projection. Since $E\leq P$, we have
${\mathcal{A}}({\mathcal{H}})E={\mathcal{T}}({\mathcal{H}})E$. Let
$\eta\in{\mathcal{H}}$ be an arbitrary vector. Choose a rank-one operator
$t\in B({\mathcal{H}})$ such that $t\xi=\eta$. Then,
$t\in{\mathcal{T}}({\mathcal{H}})$ so there is
$a\in{\mathcal{A}}({\mathcal{H}})$ with $aE=tE$. This implies that
$a\xi=aE\xi=tE\xi=\eta.$ In other words,
${\mathcal{A}}({\mathcal{H}})\xi={\mathcal{H}}$, so that $\xi$ is strictly
cyclic for ${\mathcal{A}}({\mathcal{H}})$. ∎

We can now give a complement to Corollary 5.8.

######  Theorem 5.11.

Let $q\in{\mathcal{T}}({\mathcal{H}})^{**}$ be a closed projection such that
${\mathcal{A}}({\mathcal{H}})q={\mathcal{T}}({\mathcal{H}})q$. Assume that

  1. (i)

${\mathcal{A}}({\mathcal{H}})$ admits no strictly cyclic vector in
${\mathcal{H}}$, and

  2. (ii)

there exists a non-empty ${\mathcal{M}}({\mathcal{H}})$–totally null set.

Then, $q$ is ${\mathcal{M}}({\mathcal{H}})$-totally null, and hence is
${\mathcal{A}}({\mathcal{H}})$-null.

###### Proof.

By Lemma 5.10, we see that $q\leq z_{K}$. Hence, by Lemma 5.4, there is a
closed ${\mathcal{A}}({\mathcal{H}})$-interpolation subset
$E\subset{\mathbb{S}}_{d}$ such that $\Theta(q)=\chi_{E}$. By virtue of
Theorem 5.5, to show that $q$ is ${\mathcal{M}}({\mathcal{H}})$-totally null,
it is then equivalent to show that $E$ is
${\mathcal{M}}({\mathcal{H}})$-totally null, which follows at once from
Theorem 5.7. ∎

We do not know of a characterization of those spaces ${\mathcal{H}}$ for which
${\mathcal{A}}({\mathcal{H}})$ admits no strictly cyclic vector. Nevertheless,
we can give a simple sufficient condition: such vectors cannot exist provided
that ${\mathcal{A}}({\mathcal{H}})$ is properly contained in
${\mathcal{M}}({\mathcal{H}})$ (which is the case in many classical examples).
The details are as follows. Let $f\in{\mathcal{H}}$ and
$b\in{\mathcal{M}}({\mathcal{H}})$. If $f$ is strictly cyclic, then there is
$a\in{\mathcal{A}}({\mathcal{H}})$ such that $af=bf$. This equality implies in
particular that the analytic functions $af$ and $bf$ agree on the open unit
ball. Since $f$ is non-zero, this readily implies $a=b$ using the ‘identity
theorem’ from complex analysis. Hence
${\mathcal{A}}({\mathcal{H}})={\mathcal{M}}({\mathcal{H}})$. Some additional
partial results on the existence of strictly cyclic vector can be found in
[shields1974].

##  6\. Null projections and the Riesz projection

In this section, we consider certain projections arising from convex sets of
states on a $\mathrm{C}^{*}$-algebra, and aim to understand when they are
null, or more generally when projections annihilating the ‘absolutely
continuous’ functionals are null. Let us be more precise.

Let $B$ be a $\mathrm{C}^{*}$-algebra. Let $\varphi$ and $\psi$ be continuous
linear functionals on $B$. We say that $\varphi$ is _absolutely continuous_
with respect to $\psi$ if $|\varphi|(x^{*}x)=0$ for every $x\in B^{**}$
satisfying $|\psi|(x^{*}x)=0$. Equivalently, this means that the support
projection of $|\varphi|$ is dominated by the support projection of $|\psi|$.
Here, by the support projection of a state $\theta$ on $B$, we mean the unique
projection $s_{\theta}\in B^{**}$ such that

$\\{x\in B^{**}:\theta(x^{*}x)=0\\}=B^{**}(1-s_{\theta}).$

When the support projections of $|\varphi|$ and $|\psi|$ are instead
orthogonal, we say that $\varphi$ and $\psi$ are _singular_. The reader may
consult [CT2023Henkin, Section 2] for more detail on absolute continuity and
singularity for states.

Let now $\Delta$ be a norm-closed convex set of states on $B$. We let
$\operatorname{AC}(\Delta)\subset B^{*}$ denote the set of those functionals
that are absolutely continuous with respect to some element of $\Delta$.
Similarly, we let $\operatorname{SG}(\Delta)$ denote the set of those
functionals that are mutually singular with respect to every element of
$\Delta$. By [CT2023Henkin, Theorem 3.5], there is a projection $r_{\Delta}\in
B^{**}$ with the property that
$\operatorname{AC}(\Delta)=((1-r_{\Delta})B^{**})_{\perp}$ and
$\operatorname{SG}(\Delta)=(r_{\Delta}B^{**})_{\perp}$. We sometimes refer to
$r_{\Delta}$ as the _Riesz projection_ corresponding to $\Delta$. The
following easy observation will be used several times below.

######  Lemma 6.1.

Let $\varphi,\psi\in B^{*}$ such that $\varphi$ is absolutely continuous with
respect to $\psi$. Then, the following statements hold.

  1. (i)

If $\psi\in\operatorname{AC}(\Delta)$, then
$\varphi\in\operatorname{AC}(\Delta)$.

  2. (ii)

If $\psi\in\operatorname{SG}(\Delta)$, then
$\varphi\in\operatorname{SG}(\Delta)$.

###### Proof.

(i) is trivial. To see (ii), assume that $\psi\in\operatorname{SG}(\Delta)$.
Fix $b\in B$ and put $\rho=\psi(\cdot b)$. Then,
$\rho(r_{\Delta}x)=\psi(r_{\Delta}xb)=0$ for each $x\in B^{**}$, since
$\psi\in(r_{\Delta}B^{**})_{\perp}$. This shows that
$\rho\in(r_{\Delta}B^{**})_{\perp}=\operatorname{SG}(\Delta)$. Further, the
equality $(r_{\Delta}B^{**})_{\perp}=\operatorname{SG}(\Delta)$ shows that
$\operatorname{SG}(\Delta)$ is a norm-closed subspace, so from the previous
argument we conclude that $\varphi\in\operatorname{SG}(\Delta)$ upon invoking
[CT2023Henkin, Theorem 2.7]. ∎

Let $A\subset B$ be a subalgebra. Henceforth we assume that this inclusion is
unital for simplicity. A unifying motif behind this section is to determine
when the Riesz projection $r_{\Delta}$ is $A$-null. In doing so, we will
highlight connections with many different properties of interest from the
point of view of noncommutative function theory.

First, we tackle the issue of determining when $r_{\Delta}$ lies in
$A^{\perp\perp}$. For this purpose, we need the following notion. We say that
$A$ has the _F. &M. Riesz property in $B$ with respect to $\Delta$_ if,
whenever $\varphi\in A^{\perp}$, we have that $\varphi(r_{\Delta}\cdot)\in
A^{\perp}$. In this case, we also have $\varphi((1-r_{\Delta})\cdot)\in
A^{\perp}$. In other words, this property says that the absolutely continuous
and singular parts (with respect to $\Delta$) of a functional on $B$
annihilating $A$ must also annihilate $A$. This can be reformulated as
follows.

######  Lemma 6.2.

The following statements are equivalent.

  1. (i)

$A$ has the F.&M. Riesz property in $B$ with respect to $\Delta$.

  2. (ii)

The Riesz projection $r_{\Delta}$ lies in $A^{\perp\perp}$.

###### Proof.

(i) $\Rightarrow$ (ii): Let $\varphi\in A^{\perp}$. Then,
$\varphi(r_{\Delta}\cdot)\in A^{\perp}$ by the F.&M. Riesz property.
Evaluating at $1\in A$, we obtain in particular $\varphi(r_{\Delta})=0$. We
conclude that $r_{\Delta}\in A^{\perp\perp}$.

(ii) $\Rightarrow$ (i): Let $\varphi\in A^{\perp}$. Then, $\varphi$
annihilates $A^{\perp\perp}$, and in particular it annihilates $r_{\Delta}A$
since $r_{\Delta}\in A^{\perp\perp}$. This implies in turn that
$\varphi(r_{\Delta}\cdot)$ annihilates $A$. ∎

Notice that if

(17) $A^{\perp}\subset\operatorname{AC}(\Delta)$

then trivially $A$ has the F.&M. Riesz property in $B$ with respect to
$\Delta$. We therefore refer to condition (17) as the _strong F. &M. Riesz
property in $B$ with respect to $\Delta$_ for $A$. For example, the original
F.&M. Riesz theorem shows that the disk algebra $A({\mathbb{D}})$ has the
strong F.&M. Riesz property in $\mathrm{C}({\mathbb{T}})$ with respect to
arclength measure. This can also be encoded in terms of the Riesz projection,
as we show next.

######  Lemma 6.3.

The following statements are equivalent.

  1. (i)

$A$ has the strong F.&M. Riesz property in $B$ with respect to $\Delta$.

  2. (ii)

The Riesz projection $1-r_{\Delta}$ is $A$-null.

###### Proof.

(i) $\Rightarrow$ (ii): The assumption immediately implies that
$\\{|\varphi|:\varphi\in A^{\perp}\\}\subset\operatorname{AC}(\Delta).$ Given
$\varphi\in A^{\perp}$, we thus find $|\varphi|=|\varphi|(r_{\Delta}\cdot)$
and $|\varphi|(1-r_{\Delta})=0$.

(ii) $\Rightarrow$ (i): Let $\varphi\in A^{\perp}$. By assumption,
$|\varphi|(1-r_{\Delta})=0$, so that $\varphi((1-r_{\Delta})\cdot)=0$ by Lemma
2.1. This shows that $\varphi\in\operatorname{AC}(\Delta)$. ∎

Our next result will provide several conditions that interpolate between
Lemmas 6.2 and 6.3. For this purpose, we need a technical result from
[CT2023Henkin]. A projection $q\in B^{**}$ will be said to be of _type $F$_ if
it is the supremum of an arbitrary collection of closed projections in
$B^{**}$. If the collection can be chosen to be countable, we say that $q$ is
of _type $F_{\sigma}$_. If the collection can be chosen to be finite, we say
that $q$ is of _type $F_{0}$_. If $B$ is not commutative, such a projection
need not be closed [akemann1969, Example II.6]

######  Lemma 6.4.

Assume that $\Delta$ is closed in the weak-$*$ topology of $B^{*}$. Let $\psi$
be a state in $\operatorname{SG}(\Delta)$. Then, there is a type $F_{\sigma}$
projection $q\in B^{**}$ annihilating $\operatorname{AC}(\Delta)$ and
satisfying $\psi(q)=1$.

###### Proof.

By [CT2023Henkin, Theorem 3.7], there a type $F_{\sigma}$ projection $q\in
B^{**}$ that annihilates $\Delta$ and satisfies $\psi(q)=1$. Furthermore, an
application of [CT2023Henkin, Lemma 2.4], shows that in fact $q$ annihilates
$\operatorname{AC}(\Delta)$. ∎

We say that $A$ has the _Forelli property in $B$ with respect to $\Delta$_ if
for every type $F$ projection $q\in B^{**}$ lying in
$\operatorname{AC}(\Delta)^{\perp}$, there is a contractive net $(a_{i})$ in
$A$ such that $(qa_{i})$ converges to $0$ in the weak-$*$ topology of
$B^{**}$, and $(\varphi(1-a_{i}))$ converges to $0$ for every
$\varphi\in\operatorname{AC}(\Delta)$. If this property only holds for type
$F_{\sigma}$ projections, then we say instead that $A$ has the _$\sigma$
-Forelli property in $B$ with respect to $\Delta$_. If this property only
holds for type $F_{0}$ projections, then we say instead that $A$ has the
_finite Forelli property in $B$ with respect to $\Delta$_.

These properties are inspired by a result of Forelli (see for instance
[rudin2008, Lemma 9.5.5]), which explains our choice of terminology. We now
arrive at another main result of the paper, which connects nullity of the
Riesz projection with the various properties introduced above.

######  Theorem 6.5.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
norm-closed subalgebra. Let $\Delta$ be a norm-closed convex subset of the
state space of $B$. Consider the following statements.

  1. (i)

The projection $1-r_{\Delta}$ is $A$-null.

  2. (ii)

$A$ has the strong F.&M. Riesz property in $B$ with respect to $\Delta$.

  3. (iii)

All projections in $\operatorname{AC}(\Delta)^{\perp}$ are $A$-null.

  4. (iv)

All closed projections in $\operatorname{AC}(\Delta)^{\perp}$ are $A$-null.

  5. (v)

Let $p,q\in B^{**}$ be closed projections. If
$p\in\operatorname{AC}(\Delta)^{\perp}$, then $pq\in A^{\perp\perp}$.

  6. (vi)

All closed projections in $\operatorname{AC}(\Delta)^{\perp}$ are dominated by
a projection in $\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$.

  7. (vii)

$A$ has the Forelli property in $B$ with respect to $\Delta$.

  8. (viii)

$A$ has the $\sigma$-Forelli property in $B$ with respect to $\Delta$.

  9. (ix)

$A$ has the finite Forelli property in $B$ with respect to $\Delta$.

  10. (x)

$A$ has the F.&M. Riesz property in $B$ with respect to $\Delta$.

  11. (xi)

The projection $1-r_{\Delta}$ lies in $A^{\perp\perp}$.

Then,

$\begin{array}[]{ccccccccccccccc}{\rm(i)}&\Leftrightarrow&{\rm(ii)}&\Leftrightarrow&{\rm(iii)}&\Rightarrow&{\rm(iv)}&\Leftrightarrow&{\rm(v)}&&&&&&\\\
&&&&&&&&\Downarrow&&&&&&\\\
&&&&{\rm(xi)}&\Leftrightarrow&{\rm(x)}&\Rightarrow&{\rm(vi)}&\Rightarrow&{\rm(vii)}\Rightarrow&{\rm(viii)}\Rightarrow&{\rm(ix)}\end{array}$

If we assume in addition that $\Delta$ is weak-$*$ closed in $B^{*}$, then

$\begin{array}[]{ccccccccccc}&&{\rm(i)}&\Leftrightarrow&{\rm(ii)}&\Leftrightarrow&{\rm(iii)}&\Leftrightarrow&{\rm(iv)}&\Leftrightarrow&{\rm(v)}\\\
&&&&&&&&&&\Downarrow\\\
{\rm(xi)}&\Leftrightarrow&{\rm(x)}&\Leftrightarrow&{\rm(ix)}&\Leftrightarrow&{\rm(viii)}&\Leftrightarrow&{\rm(vii)}&\Leftrightarrow&{\rm(vi)}\end{array}$

###### Proof.

(iii) $\Leftrightarrow$ (i): This is trivial since
$1-r_{\Delta}\in\operatorname{AC}(\Delta)^{\perp}$.

(i) $\Leftrightarrow$ (ii): This is Lemma 6.3.

(i) $\Rightarrow$ (iii): Any projection lying in
$\operatorname{AC}(\Delta)^{\perp}$ is dominated by $1-r_{\Delta}$, and hence
is $A$-null.

(iii) $\Rightarrow$ (iv): This is trivial.

(iv) $\Rightarrow$ (v): Let $p,q\in B^{**}$ be closed projections such that
$p\in\operatorname{AC}(\Delta)^{\perp}$. By assumption, $p$ is $A$-null, so
that $\varphi(pq)=0$ for every $\varphi\in A^{\perp}$ by Lemma 2.1. This means
that $pq\in A^{\perp\perp}$.

(v) $\Rightarrow$ (iv): Apply Proposition 2.4.

(v) $\Rightarrow$ (vi): Take $q=1$.

(vi) $\Rightarrow$ (vii): Let $q\in B^{**}$ be a type $F$ projection in
$\operatorname{AC}(\Delta)^{\perp}$. Thus, there is a collection
${\mathcal{F}}\subset B^{**}$ of closed projections such that
$q=\bigvee_{p\in{\mathcal{F}}}p$. Since
$q\in\operatorname{AC}(\Delta)^{\perp}$, we see that $q\leq 1-r_{\Delta}$.
Hence $p\leq 1-r_{\Delta}$ and so $p\in\operatorname{AC}(\Delta)^{\perp}$ for
each $p\in{\mathcal{F}}$. By assumption, for each $p\in{\mathcal{F}}$ there is
a projection $r_{p}\in\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$
such that $p\leq r_{p}$. Put $q^{\prime}=\bigvee_{p\in{\mathcal{F}}}r_{p}$,
which still lies in $A^{\perp\perp}$. Furthermore, since
$r_{p}\in\operatorname{AC}(\Delta)^{\perp}$, we find $r_{p}\leq 1-r_{\Delta}$
for each $p\in F$. Thus, $q^{\prime}\leq 1-r_{\Delta}$ and
$q^{\prime}\in\operatorname{AC}(\Delta)^{\perp}$. Now, by virtue of
Goldstine’s theorem, because $1-q^{\prime}\in A^{\perp\perp}$, we may find a
contractive net $(a_{i})$ in $A$ converging to $1-q^{\prime}$ in the weak-$*$
topology of $B^{**}$. In particular, because $q\leq q^{\prime}$, this means
that $(qa_{i})$ converges to $0$ in the weak-$*$ topology of $B^{**}$.
Further, if $\varphi\in\operatorname{AC}(\Delta)$, then since
$q^{\prime}\in\operatorname{AC}(\Delta)^{\perp}$ we have
$\varphi(q^{\prime})=0$ and

$\lim_{i}\varphi(1-a_{i})=\varphi(q^{\prime})=0.$

(vii) $\Rightarrow$ (viii) $\Rightarrow$ (ix): This is trivial.

(x) $\Leftrightarrow$ (xi): This is a direct consequence of Lemma 6.2 since
$A$ is unital.

(xi) $\Rightarrow$ (vi): If $p\in\operatorname{AC}(\Delta)^{\perp}$ is any
projection, then $p\leq 1-r_{\Delta}$, and
$1-r_{\Delta}\in\operatorname{AC}(\Delta)^{\perp}\cap A^{\perp\perp}$ by
assumption.

For the rest of the proof, we assume that $\Delta$ is closed in the weak-$*$
topology of $B^{*}$.

(ix) $\Rightarrow$ (x): Fix a bounded linear functional
$\varphi:B\to{\mathbb{C}}$ annihilating $A$. Put
$\psi=\varphi((1-r_{\Delta})\cdot)$ and $\theta=\varphi(r_{\Delta}\cdot)$, so
that $\varphi=\theta+\psi$ with $\theta\in\operatorname{AC}(\Delta)$ and
$\psi\in\operatorname{SG}(\Delta)$. The goal is to show that $\theta$ and
$\psi$ both annihilate $A$ as well, for which it clearly suffices to show that
$\theta$ annihilates $A$. Fix $a_{0}\in A$. In showing that $\theta(a_{0})=0$,
we may assume that $|\psi|$ is a state, upon scaling if necessary. Note that
$|\psi|\in\operatorname{SG}(\Delta)$ by Lemma 6.1. By Lemma 6.4, there is a
type $F_{\sigma}$ projection $r\in B^{**}$ annihilating
$\operatorname{AC}(\Delta)$ such that $|\psi|(r)=1.$ Given $\varepsilon>0$, we
may thus find a type $F_{0}$ projection $q\leq r$ such that
$|\psi|(1-q)<\varepsilon$. In particular, $q$ still annihilates
$\operatorname{AC}(\Delta)$. Applying Lemma 2.1, we obtain

(18) $|\psi((1-q)x)|<\|x\|\sqrt{\varepsilon},\quad x\in B^{**}.$

On the other hand, using the fact that $\theta\in\operatorname{AC}(\Delta)$
along with [CT2023Henkin, Lemma 2.5] we see that $\theta(\cdot
a_{0})\in\operatorname{AC}(\Delta)$ as well. Using the finite Forelli
property, we find $a\in A$ with $\|a\|\leq 1$ such that

(19) $|\theta((1-a)a_{0})|<\varepsilon$

and $|\psi(qaa_{0})|<\varepsilon$. In light of (18), this last inequality
implies

$|\psi(aa_{0})|<\|a_{0}\|\sqrt{\varepsilon}+\varepsilon.$

In turn, we have $\psi=\varphi-\theta$ and $\varphi\in A^{\perp}$, so that

(20)
$|\theta(aa_{0})|=|\psi(aa_{0})|<\|a_{0}\|\sqrt{\varepsilon}+\varepsilon.$

Combining (19) and (20), we find
$|\theta(a_{0})|<\|a_{0}\|\sqrt{\varepsilon}+2\varepsilon$. Letting
$\varepsilon\to 0$, we see that $\theta(a_{0})=0$ as desired.

(iv) $\Rightarrow$ (ii): Let $\varphi\in A^{\perp}$ and put
$\psi=\varphi((1-r_{\Delta})\cdot)$. We must show that $\psi=0$. Assume
towards a contradiction that $\psi\neq 0$. Then, $|\psi|$ lies in
$\operatorname{SG}(\Delta)$, and upon scaling if necessary, we may assume
$|\psi|$ is a state. By Lemma 6.4, there is a collection ${\mathcal{F}}\subset
B^{**}$ of closed projections such that the projection
$q=\bigvee_{p\in{\mathcal{F}}}p$ annihilates $\operatorname{AC}(\Delta)$ and
satisfies $|\psi|(q)=1$. As before in the proof of (vii), it is readily seen
that $p\in\operatorname{AC}(\Delta)^{\perp}$ for every $p\in{\mathcal{F}}$.
The assumption then shows that $p$ is $A$-null for each $p\in{\mathcal{F}}$,
and thus $q$ is $A$-null. On the other hand, we already know that
(iv)$\Rightarrow$(x)$\Leftrightarrow$(xi), so $\psi\in A^{\perp}$ and
$|\psi|(q)=0$, which is absurd. ∎

The proof of (ix) $\Rightarrow$ (x) above is inspired by that of [rudin2008,
Theorem 9.5.6].

Remark. The reader may notice that there is a notable omission in the long
list in the statement of the last theorem, namely that all closed projections
in AC$(\Delta)^{\perp}$ are in $A^{\perp\perp}$. This seems to be a very
interesting condition. It is equivalent to (iv) in the commutative case, but
we are not sure about the general case. Indeed if $B=\mathrm{C}(X)$ for some
compact Hausdorff space $X$, then this condition is simply saying that a
closed set $E$ in $X$ is a generalized $A$-peak set if each of its closed
subsets has $\mu$-measure zero for every probability measure $\mu$ in
$\Delta$. That is, a closed set $E\subset X$ is a generalized $A$-peak set if
$|\mu|(E)=0$ for every probability measure $\mu$ in $\Delta$. This clearly
implies that every closed subset of such a set $E$ is a generalized peak set,
which implies by Proposition 2.4 or the discussion above it that $\chi_{E}$ is
$A$-null. That is, $E$ is a peak-interpolation set.

When $\Delta=\\{\mu\\}$ for some regular Borel probability measure $\mu$ on
$X$, and every closed set of $\mu$-measure zero is a peak set for $A$, it
follows that every closed set of $\mu$-measure zero is $A$-null. This was
showed to us by Alexander Izzo, and it inspired several of the results in
Section 2.

##  7\. Existence of Lebesgue decompositions

In this section, we apply some of the ideas from Theorem 6.5 to prove the
existence of certain decompositions for the dual space of an operator algebra.

###  7.1. Von Neumann algebraic preliminaries

We being with some general facts about von Neumann algebras that will be
needed throughout the remainder of the paper.

Given a von Neumann algebra $M$, we let $M_{*}\subset M^{*}$ denote the norm-
closed subspace of weak-$*$ continuous linear functionals on $M$. The
following is actually a general principle for dual Banach spaces that is well
known in certain quarters. It is a necessary and sufficient condition for a
kind of ‘Kaplansky density’.

######  Lemma 7.1.

Let $M$ be a von Neumann algebra. Let $A\subset M$ be a subspace and let
$W\subset M$ denote its weak-$*$ closure. Let $\rho:M_{*}\to A^{*}$ denote the
restriction map. Then, $\rho$ has closed range if and only if there is a
constant $r>0$ such that the closed unit ball of $W$ is contained in the
weak-$*$ closure of $\\{a\in A:\|a\|\leq r\\}$.

###### Proof.

Assume first that $\rho$ has closed range. By the open mapping theorem, there
is $r>0$ such that for every $\omega\in M_{*}$, there is $\omega^{\prime}\in
M_{*}$ with $\rho(\omega)=\rho(\omega^{\prime})$ and $\|\omega^{\prime}\|\leq
r\|\rho(\omega)\|$. Then, for each $\omega\in M_{*}$ we see that

$\displaystyle\sup\\{|\omega(a)|:a\in A,\|a\|\leq r\\}$
$\displaystyle=r\|\rho(\omega)\|\geq\|\omega^{\prime}\|\geq\|\omega^{\prime}|_{W}\|=\|\omega|_{W}\|$
$\displaystyle=\sup\\{|\omega(a)|:a\in W,\|a\|\leq 1\\}.$

By the convex separation theorem, we conclude that the weak-$*$ closure of
$\\{a\in A:\|a\|\leq r\\}$ contains the closed unit ball of $W$.

Conversely, assume that there is $r>0$ such that the closed unit ball of $W$
is contained in the weak-$*$ closure of $\\{a\in A:\|a\|\leq r\\}$. Let
$\omega\in M_{*}$. It follows that

$r\|\rho(\omega)\|=\sup\\{|\omega(a)|:a\in A,\|a\|\leq
r\\}\geq\|\omega|_{W}\|.$

On the other hand, the predual of $W$ is isometrically isomorphic to
$M_{*}/W_{\perp}$, so that there is $\omega^{\prime}\in M_{*}$ such that
$\omega^{\prime}|_{W}=\omega|_{W}$ and $\|\omega^{\prime}\|\leq
2\|\omega|_{W}\|$. In particular, we find $\rho(\omega^{\prime})=\rho(\omega)$
and $\|\omega^{\prime}\|\leq 2r\|\rho(\omega)\|$, so the range of $\omega$ is
closed. ∎

There is a central projection $z\in M^{**}$ such that $M_{*}=M^{*}z$
[takesaki2002, Theorem III.2.14]. An important fact is that there is a normal
$*$-isomorphism $\Omega:M\to M^{**}z$ such that $\Omega(a)=az$ for every $a\in
M$. Next, let $\varphi$ be a normal state on $M$, and let
$\widehat{\varphi}:M^{**}\to{\mathbb{C}}$ denote its unique normal extension.
Consider the weak-$*$ closed left ideals

$L_{\varphi}=\\{a\in M:\varphi(a^{*}a)=0\\}\quad\text{and}\quad
L_{\widehat{\varphi}}=\\{x\in M^{**}:\widehat{\varphi}(x^{*}x)=0\\}.$

Then, it is readily verified that $\varphi=\widehat{\varphi}\circ\Omega$ and
that $L_{\widehat{\varphi}}=\Omega(L_{\varphi})\oplus M^{**}(1-z).$ In
particular, this gives a simple relationship between what we call the support
projection $s_{\varphi}$ of $\varphi$, and another notion of support found in
the von Neumann algebras literature. Indeed, our convention in this paper is
that $L_{\widehat{\varphi}}=M^{**}(1-s_{\varphi}).$ On the other hand, there
is a projection $e_{\varphi}\in M$ such that $L_{\varphi}=M(1-e_{\varphi})$.
It follows that $s_{\varphi}=ze_{\varphi}$. Furthermore, given another normal
state $\psi$ on $M$, we see that $L_{\varphi}\subset L_{\psi}$ if and only if
$L_{\widehat{\varphi}}\subset L_{\widehat{\psi}}$, so that our notion of
absolute continuity coincides with the corresponding notion from von Neumann
algebra theory for such states.

We say that a normal state $\varphi$ is _faithful_ if $L_{\varphi}=\\{0\\}$.
We now give an alternative description of $z$.

######  Lemma 7.2.

Let $M$ be a von Neumann algebra with a faithful normal state $\varphi$. Then,
$s_{\varphi}=z$ and $M_{*}=\operatorname{AC}(\varphi)$.

###### Proof.

Since $\varphi$ is normal, we have $\varphi(z)=1$ so $z\geq s_{\varphi}$.
Next, by the discussion above we have
$L_{\widehat{\varphi}}z=\Omega(L_{\varphi})=\\{0\\}$ since $\varphi$ is
faithful. In particular, $(1-s_{\varphi})z=0$, so that $z\leq s_{\varphi}$.
Therefore, $z=s_{\varphi}$. It follows immediately from this that
$M_{*}\subset{\rm AC}(\varphi)$. Conversely, if $\psi\in{\rm AC}(\varphi)$
then $|\psi|(1-z)=0$ since $\varphi(1-z)=0$, so indeed $\psi\in M_{*}=M^{*}z$
by Lemma 2.1. ∎

More generally, we can describe $M_{*}$ in terms of absolute continuity, as
follows. We denote by $S_{n}(M)$ the set of normal states on $M$. Assume now
that $M$ contains a weak-$*$ dense unital $\mathrm{C}^{*}$-subalgebra $B$. Let
$\rho:M^{*}\to B^{*}$ denote the restriction map. By the Kaplansky density
theorem we see that $\rho$ is isometric on $M_{*}$. In particular, this means
that the sets $(M_{*})|_{B}=\rho(M_{*})$ and $\rho(S_{n}(M))$ are norm-closed.
Put $\Delta=\rho(S_{n}(M))$; this is the norm-closed convex set in $B^{*}$
consisting of those elements of the form $\omega|_{B}$, where $\omega$ is a
normal state on $M$. We now record a simple fact.

######  Lemma 7.3.

We have that $\operatorname{AC}(S_{n}(M)))=M_{*}$ and
$\operatorname{AC}(\Delta)=(M_{*})|_{B}$.

###### Proof.

Given $\omega\in M_{*}$ and $a\in M$, we have that $\omega(\cdot a)\in M_{*}$.
Because both $M_{*}$ and $(M_{*})|_{B}$ are norm-closed, we then find
$\operatorname{AC}(S_{n}(M))\subset M_{*}$ and
$\operatorname{AC}(\Delta)\subset(M_{*})|_{B}$ by virtue of [CT2023Henkin,
Theorem 2.7]. Conversely, let $\varphi\in M_{*}$ with $\|\varphi\|=1$. We have
$\varphi(z\cdot)=\varphi$. Now, there is a partial isometry $v\in M$ such that
$|\varphi|=\varphi(\cdot v)$. It follows that $|\varphi|(z\cdot)=|\varphi|$,
whence $|\varphi|\in S_{n}(M)$ and $\varphi\in\operatorname{AC}(S_{n}(M))$. We
conclude that $M_{*}\subset\operatorname{AC}(S_{n}(M))$, which in turn plainly
implies $(M_{*})|_{B}\subset\operatorname{AC}(\Delta)$. ∎

###  7.2. Lebesgue decompositions

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
subalgebra. Given a subset $X\subset B^{*}$, we let $X|_{A}$ denote the set of
restrictions to $A$ of functionals in $X$. Following [CT2023Henkin], we say
that $X$ is a _left band_ if, whenever a functional $\varphi\in B^{*}$ is
absolutely continuous with respect to some element of $X$, then it must be
that $\varphi$ itself lies in $X$.

Next, let $M$ be a von Neumann algebra containing $B$ as a weak-$*$ dense
unital $\mathrm{C}^{*}$-subalgebra. We say that $A$ admits a _Lebesgue
decomposition_ relative to $M$ if $(M_{*})|_{A}$ is norm-closed and there is a
(necessarily unique) subspace $\Sigma\subset A^{*}$ such that
$A^{*}=(M_{*})|_{A}\oplus_{1}\Sigma$.

It follows from [CH2022, Lemma 3.4] that $B$ itself always admits a Lebesgue
decomposition relative to $M$. By contrast, the existence of such a
decomposition for $A$ is somewhat rare (see [CH2022] for more detail).
Combining Lemma 7.3 with [CH2022, Theorem 2.4], we see that if $A$ has the F.&
M. Riesz property in $B$ with respect to the convex set of restrictions to $B$
of the normal states on $M$, then $A$ admits a Lebesgue decomposition relative
to $M$ that is ‘compatible’ (in a precise sense) with that of $B$. We can now
give a different sufficient condition, where we don’t insist on this
compatibility condition.

######  Corollary 7.4.

Let $M$ be a von Neumann algebra. Let $B\subset M$ be a weak-$*$ dense unital
$\mathrm{C}^{*}$-subalgebra, and let $A\subset B$ be a unital commutative
subalgebra. Let

$X=\\{\psi\in B^{*}:\psi|_{A}\in(M_{*})|_{A}\\}.$

If $X$ is a norm-closed left band, then $A$ admits a Lebesgue decomposition
relative to $M$.

###### Proof.

Let $\Gamma\subset B^{*}$ denote the norm-closure of the convex hull of
$\\{|\psi|:\psi\in X,\|\psi\|=1\\}$. Let $r_{\Gamma}\in B^{**}$ denote the
corresponding Riesz projection [CT2023Henkin, Theorem 3.5]. By assumption, $X$
is norm-closed and a left band, so we see that $X=\operatorname{AC}(\Gamma)$.
On the other hand, it is clear that $A^{\perp}\subset X$, so in fact
$A^{\perp}\subset\operatorname{AC}(\Gamma)$. We conclude that $A$ has the
strong F.& M. Riesz property in $B$ with respect to $\Gamma$. Hence, Theorem
6.5 implies that $r_{\Gamma}\in A^{\perp\perp}$. In particular, $r_{\Gamma}$
commutes with $A$ since $A$ is assumed to be commutative. It thus follows that
$A^{*}=(A^{**}r_{\Gamma})_{\perp}\oplus_{1}(A^{**}(1-r_{\Gamma}))_{\perp}$.
Hence, it suffices to check that
$(A^{**}(1-r_{\Gamma}))_{\perp}=(M_{*})|_{A}$.

Let $\varphi\in(M_{*})|_{A}$ and choose $\omega\in M_{*}$ such that
$\omega|_{A}=\varphi$. Plainly, $\omega|_{B}\in X=\operatorname{AC}(\Gamma)$,
so by construction of the Riesz projection, we have
$\omega((1-r_{\Gamma})\cdot)=0$. Using that $r_{\Gamma}\in A^{\perp\perp}$, we
find

$\varphi((1-r_{\Gamma})a)=\omega((1-r_{\Gamma})a)=0,\quad a\in A$

so that $\varphi\in(A^{**}(1-r_{\Gamma}))_{\perp}$. Conversely let $\varphi\in
A^{*}$ such that $\varphi((1-r_{\Gamma})\cdot)=0$. By the Hahn–Banach theorem,
there is $\psi\in B^{*}$ such that $\psi|_{A}=\varphi$. Since $r_{\Gamma}\in
A^{\perp\perp}$, this implies that
$\psi(r_{\Gamma}\cdot)|_{A}=\varphi(r_{\Gamma}\cdot)=\varphi$. Put
$\psi^{\prime}=\psi(r_{\Gamma}\cdot)\in B^{*}$, which then lies in
$\operatorname{AC}(\Gamma)=X$. Hence, there is $\omega\in M_{*}$ such that
$\omega|_{A}=\psi^{\prime}|_{A}=\varphi$, so that $\varphi\in(M_{*})|_{A}$ as
desired. ∎

We now illustrate how this result applies in some classical settings.

######  Example 3.

Let $d\geq 1$ be an integer, let ${\mathbb{B}}_{d}\subset{\mathbb{C}}_{d}$ be
the open unit ball, and let ${\mathbb{S}}_{d}$ denote its topological
boundary, the unit sphere. Let $\sigma$ denote surface measure on
${\mathbb{S}}_{d}$. Let
$M=L^{\infty}({\mathbb{S}}_{d},\sigma),B=\mathrm{C}({\mathbb{S}}_{d})$ and let
$A\subset B$ denote the ball algebra, i.e. the norm-closure of the
polynomials. It is well known that the weak-$*$ closure of $A$ in $M$ is
simply $H^{\infty}({\mathbb{B}}_{d})$, by making the usual identification via
radial boundary values. Further, given $f\in H^{\infty}({\mathbb{B}}_{d})$ and
$0<r<1$, we define $f_{r}:\overline{{\mathbb{B}}_{d}}\to{\mathbb{C}}$ as
$f_{r}(z)=f(rz)$. Then, $f_{r}\in A$, $\|f_{r}\|\leq\|f\|$ and the net
$(f_{r})$ converges to $f$ in the weak-$*$ topology of $M$. We infer that the
set

$X=\\{\psi\in B^{*}:\psi|_{A}\in(M_{*})|_{A}\\}$

is norm-closed by Lemma 7.1. In turn, $X$ is a left band by Henkin’s theorem
[rudin2008, Theorem 9.3.1], Thus, $A$ admits a Lebesgue decomposition relative
to $M$ by Corollary 7.4; this also follows from [rudin2008, Section 9.8].

Next, we note that $B$ admits a Lebesgue decomposition relative to $M$ by
[CH2022, Lemma 3.4]. Nevertheless, the Lebesgue decompositions of $A$ and $B$
relative to $M$ are not compatible, by [CH2022, Theorem 4.8 and Proposition
6.3]. ∎

######  Example 4.

Let ${\mathcal{H}}$ be a Hilbert function space of the type considered in
Subsection 5.2. Let
${\mathcal{A}}({\mathcal{H}})\subset{\mathcal{T}}({\mathcal{H}})\subset
B({\mathcal{H}})$ be the operator algebras defined therein. Consider the set

$X=\\{\psi\in{\mathcal{T}}({\mathcal{H}})^{*}:\psi|_{{\mathcal{A}}({\mathcal{H}})}\in(B({\mathcal{H}})_{*})|_{{\mathcal{A}}({\mathcal{H}})}\\}.$

This is the set of ${\mathcal{M}}({\mathcal{H}})$-functionals examined in
Section 5. As explained there, this coincides with the set $\mathscr{B}$ from
[CT2023Henkin, Theorem 5.7], so we see that $X$ is a norm-closed left band.
Thus, ${\mathcal{A}}({\mathcal{H}})$ admits a Lebesgue decomposition relative
to $B({\mathcal{H}})$ by Corollary 7.4.

Next, recall that ${\mathcal{T}}({\mathcal{H}})$ contains the ideal of compact
operators on ${\mathcal{H}}$, whence it admits a Lebesgue decomposition
relative to $B({\mathcal{H}})$ by [CH2022, Lemma 3.4]. Nevertheless, the
Lebesgue decompositions of ${\mathcal{A}}({\mathcal{H}})$ and
${\mathcal{T}}({\mathcal{H}})$ relative to $B({\mathcal{H}})$ are not
compatible, by [CH2022, Theorem 4.8 and Proposition 6.1]. ∎

We emphasize again that the incompatibility of the Lebesgue decompositions in
the previous two examples shows in particular that Corollary 7.4 is applicable
in some cases where [CH2022, Theorem 2.4] is not.

##  8\. The F.& M. Riesz properties

It is a natural question whether the statements in Theorem 6.5 are ever all
equivalent, perhaps under some additional assumptions. One way of answering
this question is to determine when the F.& M. Riesz property is equivalent to
its stronger variant. We explore this issue in this section.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
subalgebra. Let $\Delta\subset B^{*}$ be a norm-closed, convex set of states
on $B$. Define

$E(\Delta)=\\{\psi\in B^{*}:\psi|_{A}\in\operatorname{AC}(\Delta)|_{A}\\}.$

This set will play an analogous role to the set of functionals whose
restrictions to $A$ are “Henkin” (see e.g. Section 8.1 below). We now show
that the F.& M. Riesz property implies a certain rigidity property for left
bands containing $\Delta$. Its relation to the the classical Cole-Range
theorem [rudin2008, Theorem 9.6.1] will be thoroughly explained at the end of
this section.

######  Theorem 8.1.

Assume that $A$ has the F.& M. Riesz property in $B$ with respect to $\Delta$.
If $X\subset B^{*}$ is a left band satisfying $\Delta\subset X\subset
E(\Delta)$, then $X=\operatorname{AC}(\Delta)$.

###### Proof.

Since $X$ is assumed to be a left band containing $\Delta$, we necessarily
have $\operatorname{AC}(\Delta)\subset X$. Conversely, fix $\varphi\in X$. We
see that $\varphi(r_{\Delta}\cdot)\in\operatorname{AC}(\Delta)\subset X$, so
it suffices to show that the functional $\psi=\varphi((1-r_{\Delta})\cdot)$ is
zero. To see this, fix $b\in B$. Since $X$ is a subspace,
$\psi=\varphi-\varphi(r_{\Delta}\cdot)\in X$. Further, since $X$ is a left
band again, we see that $\psi(\cdot b)\in X$ by [CT2023Henkin, Lemma 2.7]. By
assumption, we have $X\subset E(\Delta)$. Using the definition of $E(\Delta)$
we may find $\nu\in A^{\perp}$ and $\alpha\in\operatorname{AC}(\Delta)$ such
that $\psi(\cdot b)=\alpha+\nu$. On the other hand, $\psi(\cdot
b)\in\operatorname{SG}(\Delta)$ by Lemma 6.1. Since we have
$\nu=-\alpha+\psi(\cdot b)$, the F.& M. Riesz property implies that
$\psi(\cdot b)\in A^{\perp}$, so in particular $\psi(b)=0$. Consequently,
$\psi=0$ as desired. ∎

We can now reformulate the strong F.& M. Riesz property.

######  Corollary 8.2.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
subalgebra. Let $\Delta$ be a norm-closed convex subset of the state space of
$B$. Assume that $A$ has the F.& M. Riesz property in $B$ with respect to
$\Delta$. Then the following statements are equivalent.

  1. (i)

$A$ has the strong F.& M. Riesz property in $B$ with respect to $\Delta$.

  2. (ii)

$E(\Delta)=\operatorname{AC}(\Delta)$.

  3. (iii)

$E(\Delta)$ is a left band.

###### Proof.

(i) $\Rightarrow$ (ii): The inclusion $\operatorname{AC}(\Delta)\subset
E(\Delta)$ is trivial. Conversely, assume that $\psi\in E(\Delta)$. By
definition, this means that there is $\varphi\in\operatorname{AC}(\Delta)$
such that $\psi|_{A}=\varphi|_{A}$. Now, $\psi-\varphi\in A^{\perp}$ so by
assumption $\psi-\varphi\in\operatorname{AC}(\Delta)$, which forces
$\psi\in\operatorname{AC}(\Delta)$.

(ii) $\Rightarrow$ (iii): This is trivial.

(iii) $\Rightarrow$ (i): This follows immediately from Theorem 8.1. ∎

We remark that by using the F.& M. Riesz property, it is easy to see that the
strong F.& M. Riesz property with respect to $\Delta$ is also equivalent to
$\operatorname{SG}(\Delta)\cap A^{\perp}=\\{0\\}$.

We illustrate this result by applying it to a classical example.

######  Example 5.

Let $\lambda$ denote arclength measure on the unit circle
${\mathbb{T}}\subset{\mathbb{C}}$. By means of radial boundary values, we may
embed the algebra $H^{\infty}({\mathbb{D}})$ of bounded holomorphic functions
on the open unit disc as a unital norm-closed subalgebra of
$L^{\infty}({\mathbb{T}},\lambda)$. Choosing $\Delta=\\{\lambda\\}$, Lemma 7.2
implies that $\operatorname{AC}(\Delta)$ coincides with the weak-$*$
continuous linear functionals on $L^{\infty}({\mathbb{T}},\lambda)$. Hence,
$E(\Delta)$ coincides with the set $X$ appearing in Corollary 7.4.

Consider now the Douglas algebra
$H^{\infty}({\mathbb{D}})+\mathrm{C}({\mathbb{T}})\subset
L^{\infty}({\mathbb{T}},\lambda)$, which is known to be proper and norm-closed
(see for instance [douglas1998, Chapter 6]). Hence, there is a non-zero
bounded linear functional $\varphi$ on $L^{\infty}({\mathbb{T}},\lambda)$ that
annihilates $H^{\infty}({\mathbb{D}})+\mathrm{C}({\mathbb{T}})$. Since
$\mathrm{C}({\mathbb{T}})$ is weak-$*$ dense in
$L^{\infty}({\mathbb{T}},\lambda)$, we see that $\varphi$ is not weak-$*$
continuous, so that $H^{\infty}({\mathbb{D}})$ does not have the strong F.& M.
Riesz property in $L^{\infty}({\mathbb{T}},\lambda)$ with respect to $\Delta$.
On the other hand, it is known $H^{\infty}({\mathbb{D}})$ that has the F.& M.
Riesz property in $L^{\infty}({\mathbb{T}},\lambda)$ with respect to $\Delta$
(see [ando1978]). We may thus use Corollary 8.2 to see that the set
$E(\Delta)$ is not a left band. ∎

###  8.1. Henkin functionals and the Cole-Range phenomenon

We now wish to connect Corollary 8.2 with the main objects studied in
[CT2023Henkin]. This will be accomplished via an appropriate choice of
$\Delta$.

Let $M$ be a von Neumann algebra. Let $B\subset M$ be a unital
$\mathrm{C}^{*}$-subalgebra that is weak-$*$ dense in $M$. Let $A\subset B$ be
a unital subalgebra. A bounded linear functional $\varphi:A\to{\mathbb{C}}$ is
said to be _Henkin_ relative to $M$ if it extends weak-$*$ continuously to the
weak-$*$ closure of $A$ in $M$. This notion is in agreement with our
terminology used in Section 5. Equivalently, $\varphi$ is Henkin if it extends
weak-$*$ continuously to $M$ [rudin1991FA, Theorem 3.6], so the space of
Henkin functionals is simply $(M_{*})|_{A}$.

Let $\Delta$ denote the set of restrictions to $B$ of normal states on $M$.
Let $X\subset B^{*}$ denote the subspace consisting of functionals $\varphi\in
B^{*}$ such that $\varphi|_{A}$ is Henkin – this is precisely the set
considered in Theorem 7.4. This subspace is closed whenever $(M_{*})|_{A}$ is
closed, which can be ascertained with the help of Lemma 7.1. Further, by
virtue of Lemma 7.3, we see that $X=E(\Delta)$. In particular, Corollary 8.2
can serve as a tool for checking when $X$ forms a left band, which was a
central question in [CT2023Henkin], and shown to be useful in Corollary 7.4.

Furthermore, Theorem 8.1 can be interpreted as a unifying principle underlying
previous known results, as we illustrate next.

######  Example 6.

Assume that $X$, as defined in the last paragraph, is norm-closed (see Lemma
7.1). Clearly, $A^{\perp}\subset X$. Next, let $\Omega\subset B^{*}$ denote
the norm closure of the convex hull of $\\{|\varphi|:\varphi\in
A^{\perp},\|\varphi\|=1\\}$. In this case,
$A^{\perp}\subset\operatorname{AC}(\Omega)$. If $\psi\in E(\Omega)$ then as
before using the definition of $E(\Omega)$ we may find $\nu\in A^{\perp}$ and
$\alpha\in\operatorname{AC}(\Omega)$ such that
$\psi=\alpha+\nu\in\operatorname{AC}(\Omega)$. Thus
$E(\Omega)=\operatorname{AC}(\Omega)$. Clearly $A$ has the strong F.& M. Riesz
property in $B$ with respect to $\Omega$.

Let us now examine the assumption $\Omega\subset X\subset E(\Omega)$ from
Theorem 8.1. The inclusion $X\subset E(\Omega)$ is equivalent to
$X|_{A}\subset\operatorname{AC}(\Omega)|_{A}$, which can then be phrased in
terms of weak topologies on $A$, matching the condition of _analyticity_ in
[CT2023Henkin, Theorem 4.4], and recovering that theorem. Next, since
$A^{\perp}\subset X$, if $X$ is a left band then $\Omega\subset X$. This
follows because if $\varphi\in A^{\perp}\subset X$ then $|\varphi|\in X$, and
because $X$ is convex and assumed to be closed. Theorem 8.1 implies in turn
that $X=\operatorname{AC}(\Omega)$, assuming analyticity. This gives a variant
of [CT2023Henkin, Theorem 4.5]. ∎

######  Example 7.

We go back to the setting of Example 3, so that
$M=L^{\infty}({\mathbb{S}}_{d},\sigma)$, $B=\mathrm{C}({\mathbb{S}}_{d})$ and
$A$ is the ball algebra. Again, $X\subset B^{*}$ is the space of functionals
whose restriction to $A$ is Henkin relative to $M$. Next, let $\Omega\subset
B^{*}$ denote the norm-closed convex set of states $\rho$ such that

$\rho(a)=a(0),\quad a\in A.$

Clearly, integration against $\sigma$ is in $\Omega$ by Cauchy’s formula.
Furthermore, $\Omega\subset X$. Since Henkin functionals on $A$ admit a
weak-$*$ continuous extension to $M$, it follows that $X\subset
A^{\perp}+\operatorname{AC}(\sigma)\subset
A^{\perp}+\operatorname{AC}(\Omega)=E(\Omega)$.

It is known that $A$ has the F.& M. Riesz property in $B$ with respect to
$\Omega$ by [rudin2008, Theorem 9.5.6]. In addition, $X$ is a left band by
Henkin’s theorem [rudin2008, Theorem 9.3.1]. Hence, applying Theorem 8.1, we
find $X=\operatorname{AC}(\Omega)$, and we recover the classical
characterization of Henkin measures for the ball algebra due to Cole and Range
[rudin2008, Theorem 9.6.1]. ∎

##  9\. The Amar–Lederer property

Section 8 studied the interplay between the F.& M. Riesz properties as a
mechanism for understanding when all the statements appearing in Theorem 6.5
are equivalent. In this section, we tackle this question in different
direction, focusing instead on the implication (vi) $\Rightarrow$ (iv)
therein. As we shall see, this is a rather delicate issue.

Let $B$ be a unital $\mathrm{C}^{*}$-algebra and let $A\subset B$ be a unital
norm-closed subalgebra. Let $\Delta$ be a norm-closed convex subset of the
state space of $B$. We say that $A$ has the _Amar–Lederer property in $B$ with
respect to $\Delta$_ if all closed projections in
$\operatorname{AC}(\Delta)^{\perp}$ are dominated by an $A$-peak projection in
$\operatorname{AC}(\Delta)^{\perp}$. The reason for this terminology is made
clear in the next example, which incidentally also shows that the implication
(vi) implies (iv) generally fails in Theorem 6.5, even for commutative $B$ and
$\Delta$ a singleton, and even if we then strengthen (vi) to insist that all
projections there are closed.

######  Example 8.

Consider $H^{\infty}({\mathbb{D}})\subset L^{\infty}({\mathbb{T}},\lambda)$ as