# On various notions of distance between subalgebras of operator algebras

Ved Prakash Gupta  and  Sumit Kumar  School of Physical Sciences, Jawaharlal
Nehru University, New Delhi, INDIA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Given any irreducible inclusion $\mathcal{B}\subset\mathcal{A}$ of unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\to\mathcal{B}$, we show that the set of $E$-compatible
intermediate $C^{*}$-subalgebras is finite, thereby generalizing a finiteness
result of Ino and Watatani (from [11]). A finiteness result for a certain
collection of intermediate $C^{*}$-subalgebras of a non-irreducible inclusion
of simple unital $C^{*}$-algebras is also obtained, which provides a
$C^{*}$-version of a finiteness result of Khoshkam and Mashood (from [18]).

Apart from these finiteness results, comparisons between various notions of
distance between subalgebras of operator algebras by Kadison-Kastler,
Christensen and Mashood-Taylor are made. Further, these comparisons are used
satisfactorily to provide some concrete calculations of distance between
operator algebras associated to two distinct subgroups of a given discrete
group.

###### Key words and phrases:

Inclusions of $C^{*}$-algebras, finite-index conditional expectations,
$C^{*}$-basic construction, Hilbert $C^{*}$-modules, intermediate subalgebras,
Kadison-Kastler distance, tracial von Neumann algebras, Christensen distance,
Mashood-Taylor distance, group algebras, group $C^{*}$-algebras, group von
Neumann algebras

###### 2020 Mathematics Subject Classification:

46L05,47L40

The second named author was supported by the Council of Scientific and
Industrial Research (Government of India) through a Senior Research Fellowship
with CSIR File No. 09/0263(12012)/2021-EMR-I

##  1\. Introduction

Watatani (in [25]) (resp., Teruya and Watatani (in [23])) proved that the
lattice of intermediate subfactors of an irreducible finite-index subfactor of
type $II_{1}$ (resp., type $III$) is finite. This was then generalized to the
$C^{*}$-context by Ino and Watatani ([11]), who proved that the set of
intermediate $C^{*}$-subalgebras of an irreducible inclusion of simple unital
$C^{*}$-algebras with a finite-index conditional expectation is finite ([11,
Corollary 3.9]). Further, Longo (in [19]) obtained a bound for the cardinality
of the lattice of intermediate subfactors of any finite-index irreducible
inclusion of factors (of type $II_{1}$ or $III$). More recently, a similar
bound was obtained for the cardinality of the lattice of intermediate
$C^{*}$-algebras of a finite-index irreducible inclusion of simple unital
$C^{*}$-algebras by Bakshi and the first named author in [3], which was
achieved by introducing the notion of (interior) angle between intermediate
$C^{*}$-subalgebras. This bound was further improved by Bakshi et al. in [2].

One of the highlights of this paper is the following generalization of Ino-
Watatani’s finiteness result (from [11]):

4.2. Let $\mathcal{B}\subset{\mathcal{A}}$ be an irreducible inclusion of
unital $C^{*}$-algebras with a finite-index conditional expectation
$E:{\mathcal{A}}\to\mathcal{B}$. Then, the set
$\mathrm{IMS}(\mathcal{B},{\mathcal{A}},E)$ consisting of $E$-compatible
intermediate $C^{*}$-subalgebras of $\mathcal{B}\subset\mathcal{A}$ is finite.

Ino-Watatani’s proof of finiteness was a clever compactness argument based on
an appropriate estimate for $\|e_{\mathcal{C}}-e_{\mathcal{D}}\|$ for any two
($E$-compatible) intermediate $C^{*}$-subalgebras $\mathcal{C}$ and
$\mathcal{D}$ ([11, Lemma 3.3]), and, a perturbation result established by
them in [11, Theorem 3.8]. 4.2 is an immediate consequence of the following
more general result, whose proof is an appropriate adaptation of the
compactness argument of Ino-Watatani ([11]):

Theorem 4.1. Let $\mathcal{B}\subset{\mathcal{A}}$ be an inclusion of unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\rightarrow\mathcal{B}$. If one (equivalently, any) of the
algebras $\mathcal{C}_{\mathcal{A}}(\mathcal{B})$, $\mathcal{Z}(\mathcal{B})$
and $\mathcal{Z}(\mathcal{A})$ is finite dimensional, then the collection
$\mathcal{F}(\mathcal{B},\mathcal{A},E):=\\{\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E):\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{C}_{\mathcal{A}}(\mathcal{C})\cup\mathcal{C}\\}$
is finite.

It is noteworthy that in the above mentioned finiteness results (except
Theorem 4.1), irreducibility of the initial inclusion was crucial. Somehow,
not much is known about the finiteness of the lattice of intermediate
subalgebras of non-irreducible inclusions. Interestingly, very recently, while
looking for such finiteness results for the full lattice of intermediate von
Neumann subalgebras of non-irreducible inclusions, the compactness argument of
Ino and Watatani was also employed successfully (in [3]) by Bakshi and the
first named author to prove the following:

Theorem. [3, Theorem 6.4] Let $\mathcal{N}\subset\mathcal{M}$ be an inclusion
of von Neumann algebras with a faithful normal tracial state on $\mathcal{M}$
such that $\mathcal{Z}(\mathcal{N})$ is finite dimensional and the trace
preserving normal conditional expectation from $\mathcal{M}$ onto
$\mathcal{N}$ has finite Watatani index. If the relative commutant
$\mathcal{N}^{\prime}\cap\mathcal{M}$ equals either $\mathcal{Z}(\mathcal{N})$
or $\mathcal{Z}(\mathcal{M})$, then the lattice consisting of intermediate von
Neumann subalgebras of $\mathcal{N}\subset\mathcal{M}$ is finite.

Prior to this, Khoshkam and Mashood, in [18, Theorem 1.3], had shown that for
any finite-index subfactor $N\subset M$ of type $II_{1}$, the subcollections
$\mathcal{L}_{1}(N\subset M):=\\{P\in\mathcal{L}(N\subset M):N^{\prime}\cap
M\subset P\\}$ and $\mathcal{L}_{2}(N\subset M):=\\{P\in\mathcal{L}(N\subset
M):N^{\prime}\cap M=P^{\prime}\cap M\\}$ are both finite, where
$\mathcal{L}(N\subset M)$ denotes the lattice of intermediate subfactors of
the inclusion $N\subset M$.

Note that Theorem 4.1 (as highlighted above) is a $C^{*}$-version of [18,
Theorem 1.3] for non-irreducible inclusions of non-simple $C^{*}$-algebras.
Moreover, in Section 4 itself, we also prove another variant of a
$C^{*}$-version of [18, Theorem 1.3] for non-irreducible inclusions of simple
unital $C^{*}$-algebras, which reads as follows:

Theorem 4.7. Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of simple
unital $C^{*}$-algebras with a finite-index conditional expectation from
$\mathcal{A}$ onto $\mathcal{B}$. Then, the sublattice
$\mathcal{I}_{1}(\mathcal{B}\subset\mathcal{A}):=\\{\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A}):\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{C}\\}$
is finite.

A few words regarding the techniques employed to achieve the above mentioned
finiteness results:

Theorem 4.1 is achieved by directly employing the compactness argument of
[11], wherein the main ingredients include a basic observation from [9] (that
$e_{\mathcal{C}}\in\mathcal{A}_{1}$ for any
$\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$), an estimate for
$\|e_{\mathcal{C}}-e_{\mathcal{D}}\|$ from [11] and a perturbation result from
[8]. On the other hand, Theorem 4.7 is proved on the similar lines by first
showing that $e_{\mathcal{C}}\in\mathcal{A}_{1}$ for any intermediate
$C^{*}$-subalgebra $\mathcal{C}\in\mathcal{I}_{1}$ (Proposition 4.5), then
obtaining Ino-Watatani’s estimate for $\|e_{\mathcal{C}}-e_{\mathcal{D}}\|$
(Lemma 4.6), where $\mathcal{C}$ and $\mathcal{D}$ are two intermediate
$C^{*}$-subalgebras in $\mathcal{I}_{1}$; and, finally, by exploiting a
perturbation result by Dickson (from [8]).

The above mentioned finiteness results are all achieved in Section 4. Prior to
Section 4, we devote a short section (Section 2) on preliminaries and another
short section (Section 3) recalling and proving some basic (yet interesting)
observations related to the Kadison-Kastler distance. One interesting
observation being that for any inclusion $\mathcal{B}\subset\mathcal{A}$ of
unital $C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\to\mathcal{B}$, there exists an $\alpha>0$ such that the set of
unitaries

$\\{u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\alpha\\}\subseteq\bigcap\Big{\\{}\mathcal{N}_{\mathcal{A}}(\mathcal{C}):\mathcal{C}\in\mathcal{F}(\mathcal{B},\mathcal{A},E)\Big{\\}}$

\- see 3.7. A slightly stronger version holds for inclusions of simple unital
$C^{*}$-algebras - see 3.9. These two observations, respectively, are
immediate consequences of the above mentioned perturbation results by Ino-
Watatani and Dickson; and, a very simple minded inequality

$d_{KK}(\mathcal{B},u\mathcal{B}u^{*})\leq
2d(u,\mathcal{N}_{\mathcal{A}}(\mathcal{B})\leq 2\|u-1\|,$

obtained in Lemma 3.5, for every unitary $u$ in $\mathcal{A}$.

Then, in Section 5, we first recall and make some basic observations related
to the different notions of distance between subalgebras of $C^{*}$-algebras
and tracial von Neumann algebras, introduced by Christensen (in [5, 6, 7]) and
Mashood and Taylor (in [20]). The essence of this section lies in making
comparisons between them and the Kadison-Kastler distance; and, a desirable
(and useful) observation made in:

Proposition 5.5 Let $\mathcal{M}$ be a von Neumann algebra with a faithful
normal tracial state. Then,

$d_{MT}\big{(}P,Q\big{)}=d_{MT}\Big{(}P,\overline{Q}^{S.O.T.}\Big{)}=d_{MT}\Big{(}\overline{P}^{S.O.T.},\overline{Q}^{S.O.T.}\Big{)},$

for any two unital $*$-subalgebras $P$ and $Q$ of $\mathcal{M}$.

Finally, in Section 6, we make use of the comparisons made in Section 5 and
provide some concrete calculations of various distances between subalgebras
associated to distinct subgroups (via $C^{*}$-crossed products, Banach group
algebras and group von Neumann algebras) of a given discrete group $G$.
Interestingly, in all cases, they turn out to be distance 1 apart - see 6.6,
Proposition 6.19 and Proposition 6.20.

##  2\. Preliminaries

We briefly recall the notions of finite-index conditional expectations,
Watatani’s $C^{*}$-basic construction and compatible intermediate
$C^{*}$-subalgebras.

###  2.1. Watatani’s $C^{*}$-basic construction

For any inclusion $\mathcal{B}\subset\mathcal{A}$ of unital $C^{*}$-algebras
with common unit and a faithful conditional expectation
$E:\mathcal{A}\to\mathcal{B}$, $\mathcal{A}$ becomes a pre-Hilbert
$\mathcal{B}$-module with respect to the $\mathcal{B}$-valued inner product
$\langle\cdot,\cdot\rangle_{\mathcal{B}}:\mathcal{A}\times\mathcal{A}\to\mathcal{B}$
given by $\langle x,y\rangle_{\mathcal{B}}=E(x^{*}y)$ for $x,y\in\mathcal{A}$.
We denote by $\mathcal{E}$ the Hilbert $\mathcal{B}$-module completion of
$\mathcal{A}$.

In order to distinguish the elements of the $C^{*}$-algebra $\mathcal{A}$ and
the pre-Hilbert $\mathcal{B}$-module $\mathcal{A}$, following [24], we
consider the inclusion map $\eta:\mathcal{A}\to\mathcal{A}\subset\mathcal{E}$.
Thus,

(2.1) $\|\eta(x)\|:=\|E(x^{*}x)\|^{1/2}\leq\|x\|$

for all $x\in\mathcal{A}$. Let $\mathcal{L}_{\mathcal{B}}(\mathcal{E})$ denote
the unital $C^{*}$-algebra consisting of adjointable maps on $\mathcal{E}$.
Every member of $\mathcal{L}_{\mathcal{B}}(\mathcal{E})$ is a
$\mathcal{B}$-module map (see [24, $\S 2.10$]). Further, there is a natural
$C^{*}$-embedding
$\lambda:\mathcal{A}\to\mathcal{L}_{\mathcal{B}}(\mathcal{E})$ satisfying

$\lambda(a)\eta(x)=\eta(ax)\text{ for all }a,x\in\mathcal{A}.$

Thus, we can identify $\mathcal{A}$ with its image $\lambda(\mathcal{A})$ in
$\mathcal{L}_{\mathcal{B}}(\mathcal{E})$. Further, there is a natural
projection
$e_{1}\in\lambda(\mathcal{B})^{\prime}\cap\mathcal{L}_{\mathcal{B}}(\mathcal{E})$
(called the Jones projection corresponding to $E$) satisfying, via the above
identification, the relations

$e_{1}(\eta(x))=\eta(E(x))\text{ and }e_{1}xe_{1}=E(x)e_{1}$

for all $x\in\mathcal{A}$. The Watatani’s $C^{*}$-basic construction for the
inclusion $\mathcal{B}\subset\mathcal{A}$ with respect to the conditional
expectation $E$ is defined as the $C^{*}$-algebra

$\mathcal{A}_{1}=\overline{\mathrm{span}}\\{xe_{1}y:x,y\in\mathcal{A}\\}\subseteq\mathcal{L}_{\mathcal{B}}(\mathcal{E}).$

Like the ‘Jones’ basic construction for a subfactor, Watatani’s basic
construction also satisfies a natural universal property - see [24,
Proposition 2.2.11].

Some standard notations: Recall that for an inclusion
$\mathcal{B}\subset\mathcal{A}$ of $C^{*}$-algebras, the centralizer of
$\mathcal{B}$ in $\mathcal{A}$ is defined by

$\mathcal{C}_{\mathcal{A}}(\mathcal{B})=\\{x\in\mathcal{A}:bx=xb\text{ for all
}b\in\mathcal{B}\\};$

and, if $\mathcal{B}\subset\mathcal{A}$ are unital (with common unit), the
normalizer of $\mathcal{B}$ in $\mathcal{A}$ is defined by

$\mathcal{N}_{\mathcal{A}}(\mathcal{B})=\\{u\in\mathcal{U}(\mathcal{A}):u\mathcal{B}u^{*}=\mathcal{B}\\}.$

The centralizer $\mathcal{C}_{\mathcal{A}}(\mathcal{B})$ is a
$C^{*}$-subalgebra of $\mathcal{A}$ and is also denoted by
$\mathcal{B}^{\prime}\cap\mathcal{A}$ and called the relative commutant of
$\mathcal{B}$ in $\mathcal{A}$. The normalizer
$\mathcal{N}_{\mathcal{A}}(\mathcal{B})$ is a closed subgroup of
$\mathcal{U}(\mathcal{A})$.

###  2.2. Finite-index conditional expectations and index

For any inclusion $\mathcal{B}\subset\mathcal{A}$ of unital $C^{*}$-algebras,
a conditional expectation $E:\mathcal{A}\to\mathcal{B}$ is said to have finite
index if there exists a finite set $\\{\lambda_{i}:1\leq i\leq
n\\}\subset\mathcal{A}$ such that
$x=\sum_{i=1}^{n}E(x\lambda_{i})\lambda_{i}^{*}$ for all $x\in\mathcal{A}$.
Such a set $\\{\lambda_{i}\\}$ is called a quasi-basis for $E$ and the
Watatani index of $E$ is defined as
$\mathrm{Ind}_{W}(E)=\sum_{i=1}^{n}\lambda_{i}\lambda_{i}^{*}$, which is a
positive invertible element in $\mathcal{Z}(\mathcal{A})$ and is independent
of the quasi-basis $\\{\lambda_{i}\\}$. Every such $E$ is faithful and
preserves the unit, i.e.,
$1_{\mathcal{B}}=E(1_{\mathcal{A}})=1_{\mathcal{A}}$.

The set of finite-index conditional expectations from $\mathcal{A}$ onto
$\mathcal{B}$ is denoted by $\mathcal{E}_{0}(A,B)$. If
$\mathcal{Z}(\mathcal{A})=\mathbb{C}$ and $\mathcal{E}_{0}(A,B)\neq\emptyset$,
then an $F\in\mathcal{E}_{0}(\mathcal{A},\mathcal{B})$ is said to be minimal
if

$\mathrm{Ind}_{W}(F)=\inf\\{\mathrm{Ind}_{W}(E):E\in\mathcal{E}_{0}(\mathcal{A},\mathcal{B})\\}.$

In some nice cases, a minimal conditional expectation exists and is unique as
well.

######  Remark 2.1.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of unital $C^{*}$-algebras
with $\mathcal{E}_{0}(\mathcal{A},\mathcal{B})\neq\emptyset$ and
$\mathcal{Z}(\mathcal{A})=\mathbb{C}=\mathcal{Z}(\mathcal{B})$.

  1. (1)

There exists a unique minimal conditional expectation (denoted usually by)
$E_{0}:\mathcal{B}\to\mathcal{A}$. ([24, Theorem 2.12.3])

  2. (2)

The index of such an inclusion is defined as ([24, Definition 2.12.4])

(2.2)
$[\mathcal{A}:\mathcal{B}]_{0}=\mathrm{Ind}_{W}(E_{0})=\min\\{\mathrm{Ind}_{W}(E):E\in\mathcal{E}_{0}(\mathcal{A},\mathcal{B}).$

####  2.2.1. Compatible intermediate $C^{*}$-subalgebras

For any inclusion $\mathcal{B}\subset\mathcal{A}$ of unital $C^{*}$-algebras
(with common unit), let $\mathcal{I}(\mathcal{B}\subset\mathcal{A})$ denote
the collection of intermediate $C^{*}$-subalgebras of the inclusion
$\mathcal{B}\subset\mathcal{A}$ and let $\mathcal{E}(\mathcal{A},\mathcal{B})$
denote the set of conditional expectations from $\mathcal{A}$ onto
$\mathcal{B}$. Further, for any $E\in\mathcal{E}(\mathcal{A},\mathcal{B})$,
following [11] (also see [3, 9]), let

$\mathrm{IMS}(\mathcal{B},\mathcal{A},E):=\big{\\{}\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A}):\text{
there exists an }F\in\mathcal{E}(\mathcal{A},C)\text{ such that
}E_{\restriction_{C}}\circ F=E\big{\\}}.$

If $E$ has finite index, then for any
$\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$, a compatible
conditional expectation from $\mathcal{A}$ onto $\mathcal{C}$ is unique and
has finite index (see [11, Page 471]).

####  2.2.2. Dual conditional expectation and iterated tower of basic
constructions

######  Remark 2.2.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of unital $C^{*}$-algebras
and $E:\mathcal{A}\rightarrow\mathcal{B}$ be a finite-index conditional
expectation. Let $\mathcal{A}_{1}$ denote the $C^{*}$\- basic construction of
$\mathcal{B}\subset\mathcal{A}$ with respect to the conditional expectation
$E$ and $e_{1}$ denote the corresponding Jones projection. The following facts
are noteworthy and shall be needed ahead:

  1. (1)

$\sum_{i}\lambda_{i}e_{1}\lambda_{i}^{*}=1$ for any quasi-basis
$\\{\lambda_{i}\\}$ of $E$.

  2. (2)

$\mathcal{A}_{1}=\mathrm{span}\\{xe_{1}y:x,y\in\mathcal{A}\\}=C^{*}(\mathcal{A},e_{1})=\mathcal{L}_{\mathcal{B}}(\mathcal{A})$.
([24, Proposition 1.3.3])

  3. (3)

There exists a unique finite-index conditional expectation
$\widetilde{E}:\mathcal{A}_{1}\rightarrow\mathcal{A}$ satisfying

$\widetilde{E}(xe_{1}y)=(\mathrm{Ind}_{W}(E))^{-1}xy$

for all $x,y\in\mathcal{A}$. ([24, Proposition 1.6.1])
($\widetilde{E}$ is called the dual conditional expectation of $E$ and is
often denoted by $E_{1}$.)

  4. (4)

By iteration, one obtains a tower of unital $C^{*}$-algebras (see [13, $\S
3.1$], [15, Proposition 3.18] and [3, $\S 2.3$]))

$\mathcal{A}_{-1}:=\mathcal{B}\subset\mathcal{A}_{0}:=\mathcal{A}\subset\mathcal{A}_{1}\subset\mathcal{A}_{2}\subset\cdots\subset\mathcal{A}_{k}\subset\cdots$

with finite-index conditional expectations
$E_{k}:\mathcal{A}_{k}\to\mathcal{A}_{k-1}$ and Jones projections
$e_{k}\in\mathcal{A}_{k}$, $k\geq 1$ with
$\mathcal{A}_{k}=C^{*}(\mathcal{A}_{k-1},e_{k})$ for all $k\geq 1$. Also,
$E_{k+1}$ is the dual of $E_{k}$ for all $k\geq 0$, with $E_{0}:=E$.

  5. (5)

If $\mathrm{Ind}_{W}(E)\in B$, then
$\mathrm{Ind}_{W}(\widetilde{E})=\mathrm{Ind}_{W}(E)$. Morevoer, if
$\mathcal{A}$ and $\mathcal{B}$ are both simple and $E$ is minimal, then
$\mathcal{A}_{1}$ is simple and $\widetilde{E}$ is minimal. ([24, Cor. 2.2.14,
Prop. 2.3.4] and [15, Cor. 3.4])

  6. (6)

(Pushdown Lemma.) For each $x_{1}\in\mathcal{A}_{1}$, there exists a unique
$x_{0}\in\mathcal{A}$ such that $x_{1}e_{1}=x_{0}e_{1}$ and $x_{0}$ is given
by $x_{0}=\mathrm{Ind}_{W}(E)\widetilde{E}(x_{1}e_{1})$. ([13, Lemma 3.7])

In [3, Proposition 3.2], it was shown (using the so-called “Fourier
transforms”) that if $\mathcal{B}\subset\mathcal{A}$ is an inclusion of simple
unital $C^{*}$-algebras with
$\mathcal{E}_{0}(\mathcal{A},\mathcal{B})\neq\emptyset$, then
$\mathcal{B}^{\prime}\cap\mathcal{A}_{k}\cong\mathcal{A}^{\prime}\cap\mathcal{A}_{k+1}$
(as vector spaces) for all $k\geq 0$. In particular, if
$\mathcal{B}\subset\mathcal{A}$ is irreducible then so is
$\mathcal{A}\subset\mathcal{A}_{1}$. It turns out that the last inference is
true for more general inclusions and will be needed ahead.

######  Lemma 2.3.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of unital $C^{*}$-algebras
with a finite-index conditional expectation $E:\mathcal{A}\to\mathcal{B}$.
Then,

  1. (1)

$E_{k}(\mathcal{A}_{k-2}^{\prime}\cap\mathcal{A}_{k})=\mathcal{A}_{k-2}^{\prime}\cap\mathcal{A}_{k-1}$
for every $k\geq 1$; and,

  2. (2)

if, in addition, the inclusion $\mathcal{B}\subset\mathcal{A}$ is irreducible,
then the inclusions $\mathcal{A}_{k-1}\subset\mathcal{A}_{k}$, $k\geq 1$ are
all irreducible.

###### Proof.

It is enough to prove for $k=1$.

(1): Clearly, for each $x_{1}\in\mathcal{B}^{\prime}\cap\mathcal{A}_{1}$,
$\widetilde{E}(x_{1})b=\widetilde{E}(x_{1}b)=\widetilde{E}(bx_{1})=b\widetilde{E}(x_{1})$
for all $b\in\mathcal{B}$. Hence,
$\widetilde{E}(\mathcal{B}^{\prime}\cap\mathcal{A}_{1})\subseteq\mathcal{B}^{\prime}\cap\mathcal{A}$.

For the reverse inclusion, note that, for each
$a\in\mathcal{B}^{\prime}\cap\mathcal{A}$,
$x_{1}:=\mathrm{Ind}_{W}(E)ae_{1}\in\mathcal{B}^{\prime}\cap\mathcal{A}_{1}$
and $\widetilde{E}(x_{1})=a$, by Remark 2.2. Hence,
$\mathcal{B}^{\prime}\cap\mathcal{A}\subseteq\widetilde{E}(\mathcal{B}^{\prime}\cap\mathcal{A}_{1})$.

(2): Now, suppose that $\mathcal{B}\subset\mathcal{A}$ is irreducible and
$x_{1}\in\mathcal{A}^{\prime}\cap\mathcal{A}_{1}$. Let $\\{\lambda_{i}:1\leq
i\leq n\\}\subset\mathcal{A}$ be a quasi basis for $E$. Then,
$x_{0}:=\mathrm{Ind}_{W}(E)\widetilde{E}(x_{1}e_{1})\in\mathcal{A}$ and
$x_{1}e_{1}=x_{0}e_{1}$, by Remark 2.2(6). Further, for every $b\in B$, we
have

$x_{0}b=\mathrm{Ind}_{W}(E)\widetilde{E}(x_{1}e_{1})b=\mathrm{Ind}_{W}(E)\widetilde{E}(x_{1}e_{1}b)=\mathrm{Ind}_{W}(E)\widetilde{E}(bx_{1}e_{1})=b\mathrm{Ind}_{W}(E)\widetilde{E}(x_{1}e_{1})=bx_{0}.$

Hence, $x_{0}\in\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}=\mathbb{C}$; so that,
$x_{0}=\beta\mathbf{1}$ for some $\beta\in\mathbb{C}$, which then shows that

$x_{1}=x_{1}\sum_{i=1}^{n}\lambda_{i}e_{1}\lambda_{i}^{*}=\sum_{i=1}^{n}\lambda_{i}x_{1}e_{1}\lambda_{i}^{*}=\beta\sum_{i=1}^{n}\lambda_{i}e_{1}\lambda_{i}^{*}=\beta\mathbf{1},$

where the first equality holds because of Remark 2.2(1). This implies that
$\mathcal{A}^{{}^{\prime}}\cap\mathcal{A}_{1}=\mathbb{C}\mathbf{1}$, and we
are done. ∎

See [24, 11] for more on Watatani index, $C^{*}$-basic construction and
compatible intermediate $C^{*}$-subalgebras.

##  3\. Kadison-Kastler distance

For any normed space $X$, as is standard, its closed unit ball will be denoted
by $B_{1}(X)$ and for any subset $K$ of $X$ and an element $x\in X$, the
distance between $x$ and $K$ is defined as

$d(x,K)=\inf\\{\|x-y\|:y\in K\\}.$

######  Definition 3.1.

[14] The Kadison-Kastler distance between any two subalgebras $\mathcal{C}$
and $\mathcal{D}$ of a normed algebra $\mathcal{A}$ (which we denote by
$d_{KK}(\mathcal{C},\mathcal{D})$) is defined as the Hausdorff distance
between the closed unit balls of $\mathcal{C}$ and $\mathcal{D}$, i.e.,

$d_{KK}(\mathcal{C},\mathcal{D})=\max\left\\{\sup_{x\in
B_{1}(\mathcal{C})}d(x,B_{1}(\mathcal{D})),\sup_{z\in
B_{1}(\mathcal{D})}d(z,B_{1}(\mathcal{C}))\right\\}.$

The Kadison-Kastler distance makes sense even for two subspaces of a normed
space but we shall work mainly with the distance between subalgebras of a
normed algebra.

We must remark that the notation $d_{KK}$ is not standard. We have used it for
the Kadison-Kastler distance in order to keep a distinction between it and two
other notions of distance introduced by Christensen and a distance introduced
by Mashood and Taylor, which shall be discussed in Section 5.

Notations: For a normed algebra $\mathcal{A}$, let

$\mathrm{Sub}_{\mathcal{A}}:=\big{\\{}\text{subalgebras of
}\mathcal{A}\big{\\}};$
$C\text{-}\mathrm{Sub}_{\mathcal{A}}:=\big{\\{}\text{closed subalgebras of
}\mathcal{A}\big{\\}};$

and, if $\mathcal{A}$ is a $C^{*}$-algebra, then let

$C^{*}\text{-}\mathrm{Sub}_{\mathcal{A}}:=\big{\\{}C^{*}\text{-subalgebras of
}\mathcal{A}\big{\\}}.$

Here are some well known elementary observations related to the Kadison-
Kastler distance.

######  Remark 3.2.

Let $\mathcal{A}$ be a normed algebra.

  1. (1)

$d_{KK}(\mathcal{C},\mathcal{D})\leq 1$ for all
$\mathcal{C},\mathcal{D}\in\mathrm{Sub}_{\mathcal{A}}$.

  2. (2)

If $\mathcal{A}$ is a Banach algebra, then $d_{KK}$ is a metric on
$C\text{-}\mathrm{Sub}_{\mathcal{A}}$.

  3. (3)

If $\mathcal{C},\mathcal{D}\in C\text{-}\mathrm{Sub}_{\mathcal{A}}$ and
$\mathcal{C}$ is a proper subalgebra of $\mathcal{D}$, then
$d_{KK}(\mathcal{C},\mathcal{D})=1$. ([11, Lemma 2.1])

The following elementary observation is obvious and will be used to calculate
the distance between certain $C^{*}$-subalgebras in Section 6.

######  Lemma 3.3.

Let $\mathcal{A}$ be a normed algebra. Then,

$d_{KK}(\mathcal{C},\mathcal{D})=d_{KK}\big{(}\mathcal{C},\overline{\mathcal{D}}\big{)}=d_{KK}\big{(}\overline{\mathcal{C}},\mathcal{D}\big{)}=d_{KK}\big{(}\overline{\mathcal{C}},\overline{\mathcal{D}}\big{)}$

for all $\mathcal{C},\mathcal{D}\in\mathrm{Sub}_{\mathcal{A}}$.

It is natural to ask whether for a given subalgebra $\mathcal{B}$ of a
$C^{*}$-algebra $\mathcal{A}$, can one always find a subalgebra as close as
one desires. In [10, Example, 2.2.2], it was shown that
$d_{KK}(\mathcal{B},u\mathcal{B}u^{*})\leq\|u-\mathbf{1}\|$ for all
$u\in\mathcal{U}(\mathcal{A})$. Thus, because of the following lemma, one can
get a conjugate subalgebra as close as one wishes. (Its proof follows from an
elementary continuous functional calculus argument and we leave it to the
reader.)

######  Lemma 3.4.

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. Then, for each $\epsilon>0$,
there exists a unitary $u$ in $\mathcal{A}$ such that
$0<\|u-\mathbf{1}\|<\epsilon$.

Further, for any inclusion $\mathcal{B}\subset\mathcal{A}$ of unital
$C^{*}$-algebras, it is also quite natural to ask whether for a unitary
$u\in\mathcal{U}(A)$, there exists any relationship between its distance from
$\mathcal{N}_{\mathcal{A}}(\mathcal{B})$ and the Kadison-Kastler distance
between $\mathcal{B}$ and its conjugate $u\mathcal{B}u^{*}$.

Interestingly, motivated by an inequality given by Popa-Sinclair-Smith ([22,
Lemma 6.3]), we obtain the following pleasing relationship without much
effort, which then has a nice consequence that if
$E\in\mathcal{E}_{0}(\mathcal{A},\mathcal{B})$, then for any
$\mathcal{C}\in\mathcal{F}(\mathcal{B},\mathcal{A},E)$ (see (3.1)), a unitary
which normalizes $\mathcal{B}$ and is sufficiently close to
$\mathcal{N}_{\mathcal{A}}(\mathcal{C})$ must belong to
$\mathcal{N}_{\mathcal{A}}(\mathcal{C})$ \- see Proposition 3.6.

######  Lemma 3.5.

Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $\mathcal{B}$ be a unital
$C^{*}$-subalgebra of $\mathcal{A}$. Then,

$d_{KK}\big{(}\mathcal{B},u\mathcal{B}u^{*}\big{)}\leq
2d\big{(}u,\mathcal{N}_{\mathcal{A}}(\mathcal{B})\big{)}\leq
2\|u-\mathbf{1}\|$

for all $u\in\mathcal{U}(\mathcal{A})$.

###### Proof.

Let $u\in\mathcal{U}(\mathcal{A})$. Clearly,
$d\big{(}u,\mathcal{N}_{\mathcal{A}}(\mathcal{B})\big{)}\leq\|u-\mathbf{1}\|$.

Next, let $v\in\mathcal{N}_{\mathcal{A}}(\mathcal{B})$ and $w:=uv^{*}$. Then,
$w\mathcal{B}w^{*}=(uv^{*})\mathcal{B}(uv^{*})^{*}=u(v^{*}\mathcal{B}v)u^{*}=u\mathcal{B}u^{*}$.
So, for any $x\in B_{1}(\mathcal{B})$,

$\|x-wxw^{*}\|=\|xw-wx\|\leq\|xw-x\|+\|x-wx\|\leq 2\|w-\mathbf{1}\|.$

This implies that $d\big{(}x,B_{1}(w\mathcal{B}w^{*})\big{)}\leq
2\|w-\mathbf{1}\|.$ Hence,

$\sup_{x\in B_{1}(\mathcal{B})}d\Big{(}x,B_{1}(w\mathcal{B}w^{*})\Big{)}\leq
2\|w-\mathbf{1}\|.$

Likewise,

$\sup_{y\in B_{1}(w\mathcal{B}w^{*})}d\big{(}y,B_{1}(\mathcal{B})\big{)}\leq
2\|w-\mathbf{1}\|.$

So,

$d_{KK}\big{(}\mathcal{B},w\mathcal{B}w^{*}\big{)}\leq 2\|w-\mathbf{1}\|.$

Thus, $d_{KK}\big{(}\mathcal{B},w\mathcal{B}w^{*}\big{)}\leq 2\|u-v\|$ for all
$v\in N_{\mathcal{A}}(\mathcal{B})$. Hence,
$d_{KK}\big{(}\mathcal{B},u\mathcal{B}u^{*}\big{)}=d_{KK}\big{(}\mathcal{B},w\mathcal{B}w^{*}\big{)}\leq
2d(u,N_{\mathcal{A}}(\mathcal{B})).$ ∎

In the reverse direction, Kadison and Kastler (in [14]) had conjectured that
sufficiently close subalgebras must be conjugates of each other. This was
answered in the affirmative for various cases in a series of some fundamental
papers by Christensen, Phillips, Raeburn and others in the decade of 70s.
Since then there have been several other such so-called “perturbation
results”. People have also employed such perturbation results to answer other
important questions. One such result with a nice application is due to Ino and
Watatani (from [11]). In fact, some of the results in this article are direct
applications of the perturbation result of Ino and Watatani.

Consider the following question, which arises naturally from Lemma 3.5:

Question: Given a unital $C^{*}$-algebra $\mathcal{A}$ and a
$C^{*}$-subalgebra $\mathcal{B}$, is every unitary sufficiently close to
$\mathbf{1}$ (or to $\mathcal{N}_{\mathcal{A}}(\mathcal{B})$) in the
normalizer of the subalgebra $\mathcal{B}$?

It is easily seen - see, for instance, Example 6.17 \- that it has a negative
answer. However, interestingly, Lemma 3.5 along with a perturbation result of
Ino and Watatani ([11, Proposition 3.6]) immediately yields a somewhat
positive answer for compatible intermediate $C^{*}$-subalgebras - see
Proposition 3.6 and 3.7 below.

Notation: Given an inclusion $\mathcal{B}\subset\mathcal{A}$ of unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\to\mathcal{B}$, let

(3.1)
$\mathcal{F}(\mathcal{B},\mathcal{A},E):=\big{\\{}\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E):\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{C}_{\mathcal{A}}(\mathcal{C})\cup\mathcal{C}\big{\\}}.$

Note that
$\mathcal{F}(\mathcal{B},\mathcal{A},E)=\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$
if $\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$.

######  Proposition 3.6.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of unital $C^{*}$-algebras
with a finite-index conditional expectation $E:\mathcal{A}\to\mathcal{B}$.
Then, there exists a constant $\alpha>0$ such that
$\big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):d(u,\mathcal{N}_{\mathcal{A}}(\mathcal{C}))<\alpha\big{\\}}\subseteq\mathcal{N}_{\mathcal{A}}(\mathcal{C})$
for every $\mathcal{C}\in\mathcal{F}(\mathcal{B},\mathcal{A},E)$.

In particular, if
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$, then
$\big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):d(u,\mathcal{N}_{\mathcal{A}}(\mathcal{C}))<\alpha\big{\\}}\subseteq\mathcal{N}_{\mathcal{A}}(\mathcal{C})$
for every $\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$.

###### Proof.

Let $N$ be the number of elements (which is not unique) in a quasi-basis for
$E$ and let $\alpha=\frac{0.5}{(10N)^{4}}$. First, note that for any
$\mathcal{C}\in\mathrm{IMS}(B,A,E)$, with respect to the compatible finite-
index conditional expectation $F:\mathcal{A}\to\mathcal{C}$, and any
$u\in\mathcal{U}(\mathcal{A})$, there exists a faithful conditional
expectation $F_{u}:\mathcal{A}\to u\mathcal{C}u^{*}$ given by
$F_{u}=\mathrm{Ad}_{u}\circ F\circ\mathrm{Ad}_{u^{*}}$.

Now, let $\mathcal{C}\in\mathcal{F}(B,A,E)$ and
$u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B})$ with
$d(u,\mathcal{N}_{\mathcal{A}}(\mathcal{B}))<\alpha$. Then,
$\mathcal{B}\subseteq u\mathcal{C}u^{*}\subseteq\mathcal{A}$ and
$d_{KK}(\mathcal{C},u\mathcal{C}u^{*})<\frac{1}{(10N)^{4}}$, by Lemma 3.5.
Thus, by [11, Proposition 3.6], there exists a
$v\in\mathcal{U}(\mathcal{B}^{\prime}\cap\mathcal{A})$ such that
$v\mathcal{C}v^{*}=u\mathcal{C}u^{*}$. Since
$\mathcal{B}^{\prime}\cap\mathcal{A}\subseteq(\mathcal{C}^{\prime}\cap\mathcal{A})\cup\mathcal{C}$,
it follows that $v\mathcal{C}v^{*}=\mathcal{C}$. Hence,
$u\in\mathcal{N}_{\mathcal{A}}(\mathcal{C})$, and we are done. ∎

######  Corollary 3.7.

Let $\mathcal{A},\mathcal{B}$, $E$ and $\alpha$ be as in Proposition 3.6.
Then,

$\big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\alpha\big{\\}}\subseteq\bigcap\Big{\\{}\mathcal{N}_{\mathcal{A}}(\mathcal{C}):\mathcal{C}\in\mathcal{F}(\mathcal{B},\mathcal{A},E)\Big{\\}}.$

In particular, if
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$, then

$\big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\alpha\big{\\}}\subseteq\bigcap\Big{\\{}\mathcal{N}_{\mathcal{A}}(\mathcal{C}):\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)\Big{\\}}.$

As an application of a perturbation result by Dickson (from [8]) and a
fundamental result regarding finiteness of the index of a conditional
expectation by Izumi (from [12]), we have a slightly more general variant of
Proposition 3.6 for inclusions of simple unital $C^{*}$-algebras.

Notation: For any inclusion $\mathcal{B}\subset\mathcal{A}$ of unital
$C^{*}$-algebras (with common unit), let

(3.2)
$\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A}):=\big{\\{}\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A}):\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{C}_{\mathcal{A}}(\mathcal{C})\cup\mathcal{C}\big{\\}}.$

Clearly,
$\mathcal{F}(\mathcal{B},\mathcal{A},E)\subseteq\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A})$
for every $E\in\mathcal{E}_{0}(\mathcal{A},\mathcal{B})$; and,
$\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A})=\mathcal{I}(\mathcal{B}\subset\mathcal{A})$
if $\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$.

######  Proposition 3.8.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of simple unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\to\mathcal{B}$. Then, for any $0<\gamma<\frac{1}{10^{6}}$,
$\big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):d(u,\mathcal{N}_{\mathcal{A}}(\mathcal{C}))<\frac{\gamma}{2}\big{\\}}\subseteq\mathcal{N}_{\mathcal{A}}(\mathcal{C})$
for every $\mathcal{C}\in\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A})$.

In particular, if
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$, then
$\big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):d(u,\mathcal{N}_{\mathcal{A}}(\mathcal{C}))<\frac{\gamma}{2}\big{\\}}\subseteq\mathcal{N}_{\mathcal{A}}(\mathcal{C})$
for every $\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A})$.

###### Proof.

First, note that for any
$\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A})$, by [12,
Proposition 6.1], there exists a conditional expectation
$F:\mathcal{A}\rightarrow\mathcal{C}$ of finite index. Hence, it is faithful.

Now, let $\mathcal{C}\in\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A})$,
$0<\gamma<\frac{1}{10^{6}}$ and $u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B})$
with $d(u,\mathcal{N}_{\mathcal{A}}(\mathcal{C}))<\frac{\gamma}{2}$. Then,
$\mathcal{B}\subseteq u\mathcal{C}u^{*}\subseteq\mathcal{A}$ and
$d_{KK}(\mathcal{C},u\mathcal{C}u^{*})<\gamma<10^{-6}$, by Lemma 3.5. Note
that, as $E$ is of finite Watatani index, it satisfies the Pimsner-Popa
inequality; in particular, so does the restriction $E_{\restriction
u\mathcal{C}u^{*}}:u\mathcal{C}u^{*}\rightarrow\mathcal{B}$. Since
$\mathcal{B}$ is simple, it follows from [12, Corollary 3.4] that
$E_{\restriction u\mathcal{C}u^{*}}$ also has a finite quasi-basis. Thus, by
[8, Theorem 3.7], there exists a
$v\in\mathcal{U}(\mathcal{B}^{\prime}\cap\mathcal{A})$ such that
$v\mathcal{C}v^{*}=u\mathcal{C}u^{*}$. Since
$\mathcal{B}^{\prime}\cap\mathcal{A}\subseteq(\mathcal{C}^{\prime}\cap\mathcal{A})\cup\mathcal{C}$,
it follows that $v\mathcal{C}v^{*}=\mathcal{C}$. Hence,
$u\in\mathcal{N}_{\mathcal{A}}(\mathcal{C})$, and we are done. ∎

######  Corollary 3.9.

Let $\mathcal{A}$, $\mathcal{B}$ and $E$ be as in Proposition 3.8. Then, for
any $0<\gamma<\frac{1}{10^{6}}$,

$\Big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\frac{\gamma}{2}\Big{\\}}\subseteq\bigcap\Big{\\{}\mathcal{N}_{\mathcal{A}}(\mathcal{C}):\mathcal{C}\in\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A})\Big{\\}}.$

In particular, if
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$, then

$\Big{\\{}u\in{\mathcal{N}}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\frac{\gamma}{2}\Big{\\}}\subseteq\bigcap\Big{\\{}\mathcal{N}_{\mathcal{A}}(\mathcal{C}):\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A})\Big{\\}}.$

For any group $G$, let $\text{Sub}_{G}$ denote the collection of subgroups of
$G$. Then, $G$ admits a canonical action on $\text{Sub}_{G}$ via conjugation,
i.e.,

$G\times\text{Sub}_{G}\ni(g,H)\mapsto gHg^{-1}\in\text{Sub}_{G}.$

######  Corollary 3.10.

Let $\mathcal{A},\mathcal{B},E$ and $\alpha$ be as in Proposition 3.6 and let
$G:=\mathcal{N}_{\mathcal{A}}(\mathcal{B})$. Then, with respect to the natural
conjuation action of $G$ on $\mathrm{Sub}_{G}$, the following hold:

  1. (1)

The open ball

$\\{u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\alpha\\}\subseteq\bigcap\big{\\{}\mathrm{Stab}_{G}\big{(}\mathcal{N}_{\mathcal{A}}(\mathcal{B})\cap\mathcal{N}_{\mathcal{A}}(\mathcal{C})\big{)}:\mathcal{C}\in\mathcal{F}(\mathcal{B},\mathcal{A},E)\big{\\}};$

and, in particular, if
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$, then

$\\{u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\alpha\\}\subseteq\bigcap\big{\\{}\mathrm{Stab}_{G}\big{(}\mathcal{N}_{\mathcal{A}}(\mathcal{B})\cap\mathcal{N}_{\mathcal{A}}(\mathcal{C})\big{)}:\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)\big{\\}}.$

  2. (2)

In addition, if $\mathcal{A}$ and $\mathcal{B}$ are both simple, then for any
$0<\gamma<\frac{1}{10^{6}}$, the open ball

$\Big{\\{}u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\frac{\gamma}{2}\Big{\\}}\subseteq\bigcap\big{\\{}\mathrm{Stab}_{G}\big{(}\mathcal{N}_{\mathcal{A}}(\mathcal{B})\cap\mathcal{N}_{\mathcal{A}}(\mathcal{C})\big{)}:\mathcal{C}\in\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A})\big{\\}};$

and, in particular, if
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$, then

$\Big{\\{}u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B}):\|u-\mathbf{1}\|<\frac{\gamma}{2}\Big{\\}}\subseteq\bigcap\big{\\{}\mathrm{Stab}_{G}\big{(}\mathcal{N}_{\mathcal{A}}(\mathcal{B})\cap\mathcal{N}_{\mathcal{A}}(\mathcal{C})\big{)}:\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A})\big{\\}}.$

##  4\. Some finiteness results

This section is devoted to some more applications of certain perturbation
results from [11] and [8] to generalize some finiteness results by Ino-
Watatani ([11]) and Khoshkam-Mashood ([18]).

###  4.1. Finiteness of certain compatible intermediate $C^{*}$-subalgebras

######  Theorem 4.1.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of unital $C^{*}$-algebras
with a finite-index conditional expectation
$E:\mathcal{A}\rightarrow\mathcal{B}$. If one (equivalently, any) of the
algebras $\mathcal{C}_{\mathcal{A}}(\mathcal{B})$, $\mathcal{Z}(\mathcal{B})$
and $\mathcal{Z}(\mathcal{A})$ is finite dimensional, then the collection
$\mathcal{F}(\mathcal{B},\mathcal{A},E)$ (as in (3.1)) is finite.

###### Proof.

First, note that, since there exists a finite-index conditional expectation
from $\mathcal{A}$ onto $\mathcal{B}$, it follows from [24, Proposition 2.7.3]
that the $C^{*}$-subalgebras $\mathcal{C}_{\mathcal{A}}(\mathcal{B})$,
$\mathcal{Z}(\mathcal{B})$ and $\mathcal{Z}(\mathcal{A})$ are either all
finite dimensional or none of them is finite dimensional.

Consider the Watatani’s $C^{*}$-basic construction
$\mathcal{A}_{1}:=C^{*}(\mathcal{A},e_{1})$ of the inclusion
$\mathcal{B}\subset\mathcal{A}$ with respect to the conditional expectation
$E$ and Jones projection $e_{1}$. In view of the preceding paragraph and the
given hypothesis, $\mathcal{Z}(\mathcal{B})$ is finite dimensional; so, by
[24, Proposition 2.7.3] again, the relative commutant
$\mathcal{B}^{\prime}\cap\mathcal{A}_{1}$ is finite-dimensional. Therefore,
the set

$\mathcal{P}:=\\{p\in\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}_{1}:p\mbox{\, is
a projection }\\}$

is a compact Hausdorff space with respect to the operator norm.

Further, note that if $\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$
with respect to the compatible conditional expectation
$F:\mathcal{A}\to\mathcal{C}$, then $F$ has finite index and
$\mathcal{C}_{1}\subseteq\mathcal{A}_{1}$ (see [9, Prop. 2.7(2)]; so, the
corresponding Jones projection $e_{\mathcal{C}}$ belongs to
$\mathcal{C}^{\prime}\cap\mathcal{C}_{1}\subseteq\mathcal{B}^{\prime}\cap\mathcal{A}_{1}$,
where $\mathcal{C}_{1}$ denotes the $C^{*}$-basic construction of the
inclusion $\mathcal{C}\subset\mathcal{A}$ with respect to the finite-index
conditional expectation $F$ and Jones projection $e_{\mathcal{C}}$.

Fix a $0<\gamma<10^{-6}$ and let
$\varepsilon=\frac{\gamma}{2\,\|\mathrm{Ind}_{W}(E)\|}$. By the compactness of
$\mathcal{P}$, there exists a finite cover of $\mathcal{P}$ consisting of open
balls of radius $\varepsilon$. So, it suffices to show that each such
$\varepsilon$-ball contains only finitely many Jones projections corresponding
to the members of $\mathcal{F}(\mathcal{B},\mathcal{A},E)$.

Note that, for any two
$\mathcal{C},\mathcal{D}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$, their
corresponding Jones projections $e_{\mathcal{C}}$ and $e_{\mathcal{D}}$ are in
$\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}_{1}$ and satisfy

$d_{KK}(\mathcal{C},\mathcal{D})\leq\|\mathrm{Ind}_{W}(E)\|\|e_{\mathcal{C}}-e_{\mathcal{D}}\|,$

by [11, Lemma 3.3]. In particular, if $e_{\mathcal{C}}$ and $e_{\mathcal{D}}$
are in one of the $\varepsilon$-open balls, then
$d_{KK}(\mathcal{C},\mathcal{D})\leq\frac{\gamma}{2}<\gamma$. Thus, by [8,
Theorem 3.7], there exists a unitary $u$ in
$\mathcal{B}^{\prime}\cap\mathcal{A}$ such that
$\mathcal{C}=u\mathcal{D}u^{*}$ (and $\|u-1\|\leq
16\sqrt{110}\gamma^{\frac{1}{2}}+880\gamma$).

Let $\mathcal{C},\mathcal{D}\in\mathcal{F}(\mathcal{B},\mathcal{A},E)$ and
they be in one $\varepsilon$-ball as above. As
$u\in\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}$ and
$\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}\subseteq\mathcal{C}_{\mathcal{A}}(\mathcal{D})\cup\mathcal{D}$,
it follows that $\mathcal{C}=u\mathcal{D}u^{*}=\mathcal{D}$. Thus, each
$\varepsilon$-open ball of the cover contains at most one Jones projection for
some member of $\mathcal{F}(\mathcal{B},\mathcal{A},E)$, as was desired. ∎

We can then immediately deduce the following generalization of [11, Corollary
3.9]. (The second part follows from Lemma 2.3.)

######  Corollary 4.2.

Let $\mathcal{B}\subset\mathcal{A}$ be an irreducible inclusion of unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\to\mathcal{B}$. Then, $\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$
is finite.

In particular, $\mathrm{IMS}(\mathcal{A}_{k},\mathcal{A}_{k+1},E_{k+1})$ is
finite for every $k\geq 0$.

For any unital inclusion of von Neumann algebras
$\mathcal{N}\subset\mathcal{M}$, let
$\mathcal{L}(\mathcal{N}\subset\mathcal{M})$ denote the lattice of
intermediate von Neumann subalgebras. For any such inclusion, if there exists
a faithful normal tracial state $\mathrm{tr}$ on $\mathcal{M}$ and the unique
$\mathrm{tr}$-preserving conditional expectation
$E_{\mathcal{N}}:\mathcal{M}\to\mathcal{N}$ has finite Watatani index, then it
was shown in [3] that if $\mathcal{Z}(\mathcal{N})$ is finite-dimensional and
the relative commutant $\mathcal{N}^{\prime}\cap\mathcal{M}$ equals either
$\mathcal{Z}(\mathcal{N})$ or $\mathcal{Z}(\mathcal{M})$, then
$\mathcal{L}(\mathcal{N}\subset\mathcal{M})$ is finite. Analogous to [18,
Theorem 1.3] and Theorem 4.1, we now have the following:

######  Proposition 4.3.

Let $\mathcal{N}\subset\mathcal{M}$ be a unital inclusion of finite von
Neumann algebras with a (fixed) faithful normal tracial state $\mathrm{tr}$ on
$\mathcal{M}$ such that the unique $\mathrm{tr}$-preserving conditional
expectation $E_{\mathcal{N}}:\mathcal{M}\to\mathcal{N}$ has finite Watatani
index. If one (equivalently, any) of the algebras
$\mathcal{C}_{\mathcal{M}}(\mathcal{N})$, $\mathcal{Z}(\mathcal{N})$ and
$\mathcal{Z}(\mathcal{M})$ is finite dimensional, then the subcollection

$\mathcal{L}_{0}(\mathcal{N}\subset\mathcal{M}):=\\{\mathcal{P}\in\mathcal{L}(\mathcal{N}\subset\mathcal{M}):\mathcal{N}{{}^{\prime}}\cap\mathcal{M}\subseteq\mathcal{P}\cup(\mathcal{P}{{}^{\prime}}\cap\mathcal{M})\\}$

is finite.

###### Proof.

Clearly,
$\mathcal{L}(\mathcal{N}\subset\mathcal{M})\subseteq\mathrm{IMS}(\mathcal{N},\mathcal{M},E_{\mathcal{N}})$.
Rest follows from Theorem 4.1. ∎

Note that, Proposition 4.3 also generalizes Watatani’s finiteness result [25,
Theorem 2.2] to a non-irreducible setting.

###  4.2. Finiteness results for non-irreducible inclusions of simple
$C^{*}$-algebras

In general, if $\mathcal{B}\subset\mathcal{A}$ are simple unital
$C^{*}$-algebras and the inclusion is not irreducible, then the lattice
$\mathcal{I}(\mathcal{B}\subset\mathcal{A})$ need not be finite. For instance,
consider the following easy example:

######  Example 4.4.

Let $\mathcal{A}=M_{2}(\mathbb{C})$, $\mathcal{B}=\mathbb{C}I_{2}$,
$\Delta=\\{\mathrm{diag}(\lambda,\mu)\,\,:\,\,\lambda,\,\mu\,\in\mathbb{C}\\}$,
$E:\mathcal{A}\rightarrow\mathcal{B}$ denote the canonical (tracial)
conditional expectation given by

$E([a_{ij}])=\frac{(a_{11}+a_{22})}{2}I_{2},\ \ [a_{ij}]\in\mathcal{A};$

and, let $F:\mathcal{A}\rightarrow\Delta$ denote another canonical conditional
expectation given by $F([a_{ij}])=\mathrm{diag}(a_{11},a_{22})$,
$[a_{ij}]\in\mathcal{A}$.

Note that $u\Delta u^{*}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ for all
$u\in U(2)$ (by [9, Lemma 2.8]). Also, the set $\\{u\Delta u^{*}:u\in
U(2)\setminus\mathcal{N}_{\mathcal{A}}(\Delta)\\}$ is infinite because the set
of left cosets of $\mathcal{N}_{\mathcal{A}}(\Delta)$ in $U(2)$ is infinite.
Hence, $\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ and, therefore,
$\mathcal{I}(\mathcal{B}\subset\mathcal{A})$ are infinite sets.

Here is an indirect way of seeing why $U(2)/\mathcal{N}_{\mathcal{A}}(\Delta)$
is infinite. Suppose, on contrary, that
$\\{[u]:=u\mathcal{N}_{\mathcal{A}}(\Delta):u\in U(2)\\}$ is finite. Then, for
any element $w\in u\mathcal{N}_{\mathcal{A}}(\Delta)$, $w=uv$ for some
$v\in\mathcal{N}_{\mathcal{A}}(\Delta)$; so,

$\alpha(\Delta,w\Delta
w^{*})=\alpha(\Delta,uv\Delta(uv)^{*})=\alpha(\Delta,u\Delta u^{*}),$

where $\alpha$ is the interior angle (see [9]). This implies that
$\alpha(\Delta,w\Delta w^{*})=\alpha(\Delta,u\Delta u^{*})$ for every
$w\in[u]$. Hence, the set

$\\{\alpha(\Delta,u\Delta u^{*}):u\in U(2)\\}=\left\\{\alpha(\Delta,u\Delta
u^{*})\,\,:\,\,[u]\in\frac{U(2)}{\mathcal{N}_{\mathcal{A}}(\Delta)}\right\\}$

is finite. This contradicts the fact that $\\{\alpha(\Delta,u\Delta
u^{*}):u\in U(2)\\}=[0,\frac{\pi}{2}]$ (see [9, Corollary 4.6]). Thus, the set
of left cosets of $\mathcal{N}_{\mathcal{A}}(\Delta)$ in $U(2)$ must be
infinite.

However, for a non-irreducible inclusion $\mathcal{B}\subset A$ of simple
unital $C^{*}$-algebras with a finite-index conditional expectation, we shall
show in this section that the sublattice consisting of intermediate
$C^{*}$-subalgebras of $\mathcal{B}\subset\mathcal{A}$ which contain the
centralizer algebra $\mathcal{C}_{\mathcal{A}}(\mathcal{B})$ is finite.

The following useful observation (whose first part comes from [12] and the
second part is comparable with [9, Proposition 2.7 (1, 2)]) will be needed
ahead.

######  Proposition 4.5.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of simple unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\rightarrow\mathcal{B}$. Then, for any
$\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A})$, there exists a
finite-index conditional expectation $F:\mathcal{A}\to\mathcal{C}$ and,
moreover, if $\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{C}$,
then
$\mathcal{L}_{\mathcal{C}}(\mathcal{A})\subseteq\mathcal{L}_{\mathcal{B}}(\mathcal{A}),$
where $\mathcal{L}_{\mathcal{C}}(\mathcal{A})$ is defined with respect to $F$.

In particular, $\mathcal{C}_{1}\subset\mathcal{A}_{1}$, where
$\mathcal{A}_{1}$ (resp., $\mathcal{C}_{1}$) denotes the $C^{*}$-basic
construction of the inclusion $\mathcal{B}\subset\mathcal{A}$ (resp.,
$\mathcal{C}\subset\mathcal{A}$) with respect to $E$ (resp., $F$).

###### Proof.

Let $\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A})$. Then, by [12,
Proposition 6.1], there exists a conditional expectation
$F:\mathcal{A}\rightarrow\mathcal{C}$ of finite index.

Next, suppose that
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})=\mathcal{B}^{\prime}\cap\mathcal{A}\subseteq\mathcal{C}$.

Note that, as $E$ is of finite Watatani index, it satisfies the Pimsner-Popa
inequality ([24]); in particular, so does the restriction $E_{\restriction
C}:\mathcal{C}\rightarrow\mathcal{B}$. Since $\mathcal{B}$ is simple, it then
follows from [12, Corollary 3.4] that $E_{\restriction_{\mathcal{C}}}$ also
has a finite quasi-basis. Thus, $G:=E_{\restriction\mathcal{C}}\circ F$ is of
finite Watatani index by [24, Proposition 1.7.1]. Since $E$ and $G$ are two
finite-index conditional expectations from $\mathcal{A}$ onto $\mathcal{B}$,
by [24, Proposition 2.10.9], there exists a
$q\in\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}$ such that $E(x)=G(q^{*}xq)\
\mbox{for all }x\in\mathcal{A}.$

Then, for any $T\in\mathcal{L}_{C}(\mathcal{A})$, we observe that

$\displaystyle\langle T(x),y\rangle_{\mathcal{B}}$ $\displaystyle=$
$\displaystyle E(T(x)^{*}y)$ $\displaystyle=$ $\displaystyle
G(q^{*}T(x)^{*}yq)$ $\displaystyle=$ $\displaystyle
E_{\restriction_{C}}(F((T(x)q)^{*}yq))$ $\displaystyle=$ $\displaystyle
E_{\restriction_{C}}(\langle T(x)q,yq\rangle_{C})$ $\displaystyle=$
$\displaystyle E_{\restriction C}(\langle xq,T^{*}(yq)\rangle_{C})$
$\displaystyle=$ $\displaystyle E_{\restriction
C}\big{(}F(q^{*}x^{*}T^{*}(yq))\big{)}$ $\displaystyle=$ $\displaystyle
E_{\restriction C}\big{(}F(q^{*}x^{*}T^{*}(y)q)\big{)}$ $\displaystyle=$
$\displaystyle G(q^{*}x^{*}T^{*}(y)q)$ $\displaystyle=$ $\displaystyle
E(x^{*}T^{*}(y))$ $\displaystyle=$ $\displaystyle\langle
x,T^{*}(y)\rangle_{\mathcal{B}}$

for all $x,y\in\mathcal{A}$. Hence,
$T\in\mathcal{L}_{\mathcal{B}}(\mathcal{A})$. ∎

Notation: For any inclusion $\mathcal{B}\subset\mathcal{A}$ of unital
$C^{*}$-algebras, let

$\mathcal{I}_{1}(\mathcal{B}\subset\mathcal{A}):=\\{\mathcal{C}\in\mathcal{I}(\mathcal{B}\subset\mathcal{A}):\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{C}\\}.$

Clearly, $\mathcal{I}_{1}$ a sublattice of
$\mathcal{I}(\mathcal{B}\subset\mathcal{A})$,
$\mathcal{I}_{1}(\mathcal{B}\subset\mathcal{A})\subseteq\mathcal{I}_{0}(\mathcal{B}\subset\mathcal{A})$
and
$\mathcal{I}_{1}(\mathcal{B}\subset\mathcal{A})=\mathcal{I}(\mathcal{B}\subset\mathcal{A})$
if $\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$.

We shall need the following adaptation of [11, Lemma 3.3].

######  Lemma 4.6.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of simple unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\rightarrow\mathcal{B}$. Then,

(4.1)
$d_{KK}(\mathcal{C},\mathcal{D})\leq\mathrm{Ind}_{W}(E)\,\|e_{\mathcal{C}}-e_{\mathcal{D}}\|$

for all
$\mathcal{C},\mathcal{D}\in\mathcal{I}_{1}(\mathcal{B}\subset\mathcal{A})$.

###### Proof.

We shall simply write $\mathcal{I}_{1}$ for the set
$\mathcal{I}_{1}(\mathcal{B}\subset\mathcal{A})$.

Since $E$ has finite index, it satisfies the Pimsner-Popa inequality (by [24,
Proposition 2.6.2]), i.e., $E(x^{*}x)\geq(\mathrm{Ind}_{W}(E))^{-1}x^{*}x$ for
all $x\in\mathcal{A}$; and, since $\mathrm{Ind}_{W}(E)\geq 1$ (by [24, Lemma
2.3.1]), it follows that

$E(x^{*}x)\geq(\mathrm{Ind}_{W}(E))^{-2}x^{*}x$

for all $x\in\mathcal{A}$. In particular,
$\|\eta(x)\|\geq(\mathrm{Ind}_{W}(E))^{-1}\|x\|$ for all $x\in\mathcal{A}$.
Moreover, $\|\eta(a)\|\leq\|a\|\leq 1$ for every $a\in B_{1}(\mathcal{C})$.

Since $\mathcal{B}\subset\mathcal{A}$ are both simple, it follows from
Proposition 4.5 that
$\mathcal{L}_{\mathcal{C}}(\mathcal{A})\cup\mathcal{L}_{\mathcal{D}}(\mathcal{A})\subseteq\mathcal{L}_{\mathcal{B}}(\mathcal{A})$,
for any two $\mathcal{C},\mathcal{D}\in\mathcal{I}_{1}$. Thus,
$e_{\mathcal{C}},e_{\mathcal{D}}\in\mathcal{L}_{\mathcal{B}}(\mathcal{A})$,
and

$\|e_{\mathcal{C}}-e_{\mathcal{D}}\|\geq\frac{\|\eta(E_{\mathcal{C}}(a)-E_{\mathcal{D}}(a))\|}{\|\eta(a)\|}\geq\|\eta(a-E_{\mathcal{D}}(a))\|\geq\frac{1}{\mathrm{Ind}_{W}(E)}\|a-E_{\mathcal{D}}(a)\|$

for all $a\in B_{1}(\mathcal{C})\setminus\\{0\\}$. Therefore, for each $a\in
B_{1}(\mathcal{C})$, there exists an element $b:=E_{\mathcal{D}}(a)\in
B_{1}(\mathcal{D})$ such that
$\|a-b\|\leq\mathrm{Ind}_{W}(E)\,\|e_{\mathcal{C}}-e_{\mathcal{D}}\|$. By a
symmetric arguement, we deduce that

$d_{KK}(\mathcal{C},\mathcal{D})\leq\mathrm{Ind}_{W}(E)\,\|e_{\mathcal{C}}-e_{\mathcal{D}}\|.$

∎

The following finiteness result is an adaptation (as well as a mild
generalization) of [11, Corollary 3.9] (also see [18, Theorem 1.3]). Its proof
is an imitation of that of Theorem 4.1. However, we provide all steps for the
sake of completeness.

######  Theorem 4.7.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of simple unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\rightarrow\mathcal{B}$. Then, the sublattice
$\mathcal{I}_{1}(\mathcal{B}\subset\mathcal{A})$ is finite.

###### Proof.

Consider the Watatani’s $C^{*}$-basic construction
$\mathcal{A}_{1}:=C^{*}(\mathcal{A},e_{1})$ of the inclusion
$\mathcal{B}\subset\mathcal{A}$ with respect to the finite-index conditional
expectation $E$ and Jones projection $e_{1}$. Let
$\widetilde{E}:\mathcal{A}_{1}\rightarrow\mathcal{A}$ be the finite-index dual
conditional expectation of $E$. Then,
$\widetilde{E}\circ{E}:\mathcal{A}_{1}\rightarrow\mathcal{B}$ is also a
conditional expectation of finite index. Since $\mathcal{A}$ is simple, it
follows from [24, Proposition 2.7.3] that the relative commutant
$\mathcal{B}^{\prime}\cap\mathcal{A}_{1}$ is finite-dimensional. Therefore,
the set

$\mathcal{P}:=\\{p\in\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}_{1}:p\mbox{\, is
a projection}\\}$

is a compact Hausdorff space with respect to the operator norm.

Since $\mathcal{A}$ is simple, $\mathrm{Ind}_{W}(E)$ is a positive scalar. Fix
a $0<\gamma<10^{-6}$ and let
$\varepsilon=\frac{\gamma}{2\,\mathrm{Ind}_{W}(E)}$. By the compactness of
$\mathcal{P}$, there exists a finite cover of $\mathcal{P}$ consisting of open
balls of radius $\epsilon$. Now, it suffices to show that each such
$\epsilon$-ball contains only finitely many Jones projections corresponding to
members of $\mathcal{I}_{1}$.

Note that, for any two $\mathcal{C},\mathcal{D}\in\mathcal{I}_{1}$, their
corresponding Jones projections $e_{\mathcal{C}}$ and $e_{\mathcal{D}}$ are in
$\mathcal{B}^{{}^{\prime}}\cap\mathcal{L}_{\mathcal{B}}(\mathcal{A})=\mathcal{B}^{{}^{\prime}}\cap\mathcal{A}_{1}$,
by Proposition 4.5; and, by Lemma 4.6, they also satisfy the inequality

$d_{KK}(\mathcal{C},\mathcal{D})\leq\mathrm{Ind}_{W}(E)\|e_{\mathcal{C}}-e_{\mathcal{D}}\|.$

So, if $e_{\mathcal{C}}$ and $e_{\mathcal{D}}$ are in one of the
$\varepsilon$-open balls, then
$d_{KK}(\mathcal{C},\mathcal{D})\leq\frac{\gamma}{2}<\gamma$. Thus, by [8,
Theorem 3.7], there exists a unitary $u$ in
$\mathcal{B}^{\prime}\cap\mathcal{A}$ such that
$\mathcal{D}=u\mathcal{C}u^{*}$ (and $\|u-1_{\mathcal{A}}\|\leq
16\sqrt{110}\gamma^{\frac{1}{2}}+880\gamma$). Since
$\mathcal{B}^{\prime}\cap\mathcal{A}=\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{C}$,
this implies that $u\in\mathcal{C}$. Hence,
$\mathcal{D}=u\mathcal{C}u^{*}=\mathcal{C}$. This shows that each
$\varepsilon$-open ball of the cover contains at most one Jones projection for
some intermediate $C^{*}$-subalgebra in $\mathcal{I}_{1}$, and we are done. ∎

Since countably decomposable type $III$ factors are simple, we obtain the
following generalization of [23, Theorem 2.5] (and a partial generalization of
[18, Theorem 1.3]).

######  Corollary 4.8.

Let $\mathcal{N}\subset\mathcal{M}$ be an inclusion of countably decomposable
type $III$ factors with a finite-index conditional expectation. Then, the set
$\mathcal{L}_{1}(\mathcal{N}\subset\mathcal{M}):=\\{\mathcal{Q}\in\mathcal{L}(\mathcal{N}\subset\mathcal{M}):\mathcal{N}^{\prime}\cap\mathcal{M}\subseteq\mathcal{Q}\\}$
is finite.

###  4.3. Cardinality of $\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$

Longo (in [19]) showed that the cardinality of the lattice of intermediate
subfactors of any finite-index irreducible inclusion $N\subset M$ of factors
(of type $II_{1}$ or $III$) is bounded by $([M:N]^{2})^{[M:N]^{2}}$. More
recently, a similar bound was obtained for the lattice of intermediate
$C^{*}$-algebras of a finite-index inclusion of simple unital $C^{*}$-algebras
by Bakshi and the first named author in [3], which was achieved by introducing
the notion of (interior) angle between two intermediate $C^{*}$-subalgebras.
Further, Bakshi et al (in [2]) improved the bound by proving that the
cardinality of the lattice of intermediate $C^{*}$-subalgebras of an
irreducible inclusion $\mathcal{B}\subset\mathcal{A}$ of simple unital
$C^{*}$-algebras with finite index is bounded by
$9^{[\mathcal{A}:\mathcal{B}]_{0}}$.

In this subsection, for any irreducible inclusion
$\mathcal{B}\subset\mathcal{A}$ of unital $C^{*}$-algebras with a finite index
conditional expectation $E:\mathcal{A}\to\mathcal{B}$, through some elementary
group action technique, we provide an expression for the cardinality of
$\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ in terms of the indices of the
unitary normalizers of its members in $\mathcal{N}_{\mathcal{A}}(\mathcal{B})$
\- see Proposition 4.10. On face, the expression looks a bit dry, but it might
prove useful while performing some concrete calculations.

In [9, Lemma 2.8(2)], it was shown that for any inclusion
$\mathcal{B}\subset\mathcal{A}$ of unital $C^{*}$-algebras with a tracial
finite-index conditional expectation $E:\mathcal{A}\to B$,
$u\mathcal{C}u^{*}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ for every
$\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ and
$u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B})$. We now have its following (more
obvious) variant (which does not require $E$ to be tracial), which gives an
abundance of compatible intermediate $C^{*}$-algebras and will also allow us
to get an expression for the cardinality of
$\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ when the inclusion
$\mathcal{B}\subset\mathcal{A}$ is irreducible.

######  Lemma 4.9.

Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of unital $C^{*}$-algebras
with a finite-index conditional expectation
$E:\mathcal{A}\rightarrow\mathcal{B}$ such that
$\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$. Let
$\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ with respect to the
compatible finite-index conditional expectation
$F:\mathcal{A}\rightarrow\mathcal{C}$. Then, for every
$u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B})$,
$u\mathcal{C}u^{*}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ with respect to
the conditional expectation $F_{u}:\mathcal{A}\rightarrow u\mathcal{C}u^{*}$
given by $F_{u}=\mathrm{Ad}_{u}\circ F\circ\mathrm{Ad}_{u^{*}}$.

###### Proof.

As $\mathcal{C}_{\mathcal{A}}(\mathcal{B})\subseteq\mathcal{B}$ and
$\mathcal{E}_{0}(\mathcal{A},\mathcal{B})\neq\emptyset$, it follows from [24,
Corollary 1.4.3] that $E$ is the unique conditional from $\mathcal{A}$ onto
$\mathcal{B}$. Also, from [9, Lemma 2.8(1)] we know that $F_{u}$ is of finite
index. So, $E_{\restriction_{u\mathcal{C}u^{*}}}\circ F_{u}$ is again a
finite-index conditional expectation from $\mathcal{A}$ onto $\mathcal{B}$.
Thus, by uniqueness of $E$, we get $E_{\restriction_{u\mathcal{C}u^{*}}}\circ
F_{u}=E$. Hence, $u\mathcal{C}u^{*}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$
with respect to the compatible conditional expectation $F_{u}$. ∎

With notations as in Lemma 4.9, we thus observe that the group
$G:=\mathcal{N}_{\mathcal{A}}(\mathcal{B})$ admits an action on the set
$\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ via the map

$G\times\mathrm{IMS}(\mathcal{B},\mathcal{A},E)\ni(u,\mathcal{C})\mapsto
u\mathcal{C}u^{*}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E).$

For any $\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$, its
stabilizer

$\mathrm{Stab}_{G}(\mathcal{C})=\mathcal{N}_{\mathcal{A}}(\mathcal{B})\cap\mathcal{N}_{\mathcal{A}}(\mathcal{C}).$

Let $\widehat{\mathrm{IMS}(\mathcal{B},\mathcal{A},E)}$ denote a set of
representatives of the orbits of $\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$
with respect to this action. Then, in view of 4.2, from some basic theory of
group actions, we immediately conclude the following:

######  Proposition 4.10.

Let $\mathcal{B}\subset\mathcal{A}$ be an irreducible inclusion of unital
$C^{*}$-algebras with a finite-index conditional expectation
$E:\mathcal{A}\rightarrow\mathcal{B}$ and
$G:=\mathcal{N}_{\mathcal{A}}(\mathcal{B})$. Then,

  1. (1)

$\big{[}\mathcal{N}_{\mathcal{A}}(\mathcal{B}):\mathcal{N}_{\mathcal{A}}(\mathcal{B})\cap\mathcal{N}_{\mathcal{A}}(\mathcal{C})\big{]}<\infty$
for every $C\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$; and,

  2. (2)

$\big{|}\mathrm{IMS}(\mathcal{B},\mathcal{A},E)\big{|}=\sum_{\mathcal{C}\in\widehat{\mathrm{IMS}(\mathcal{B},\mathcal{A},E)}}\big{[}\mathcal{N}_{\mathcal{A}}(\mathcal{B}):\mathcal{N}_{\mathcal{A}}(\mathcal{B})\cap\mathcal{N}_{\mathcal{A}}(\mathcal{C})\big{]}.$

######  Lemma 4.11.

Let $\mathcal{A},\mathcal{B}$, $E$ and $G$ be as in Proposition 4.10 and
$\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$. Then,

$[\mathcal{A}:\mathcal{D}]_{0}=[\mathcal{A}:\mathcal{C}]_{0}\ \text{and}\
[\mathcal{D}:\mathcal{B}]_{0}=[\mathcal{C}:\mathcal{B}]_{0}$

for very $\mathcal{D}\in\mathrm{Orb}_{G}(\mathcal{C})$.

###### Proof.

Since $\mathcal{B}\subset\mathcal{A}$ is an irreducible inclusion, the
conditional expectation $E:\mathcal{A}\rightarrow\mathcal{B}$ is unique, by
[24, Corollary 1.4.3]. Hence, it is minimal.

Suppose that $\mathcal{C}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ with
respect to the compatible finite-index conditional expectation
$F:\mathcal{A}\rightarrow C$. Note that
$\mathrm{Orb}_{G}(\mathcal{C})=\\{u\mathcal{C}u^{*}:u\in G\\}$.

Let $\mathcal{D}\in\mathrm{Orb}_{G}(\mathcal{C})$. Then,
$\mathcal{D}=u\mathcal{C}u^{*}$ for some
$u\in\mathcal{N}_{\mathcal{A}}(\mathcal{B})$ and
$\mathcal{D}\in\mathrm{IMS}(\mathcal{B},\mathcal{A},E)$ with respect to the
conditional expectation $F_{u}:=\mathrm{Ad}_{u}\circ
F\circ\mathrm{Ad}_{u^{*}}$. Also,
$\mathcal{Z}(\mathcal{D})=\mathcal{Z}(u\mathcal{C}u^{*})=u\mathcal{Z}(\mathcal{C})u^{*}=\mathbb{C}=\mathcal{Z}(\mathcal{A})$,
$\mathcal{D}{{}^{\prime}}\cap\mathcal{A}\subseteq\mathcal{B}{{}^{\prime}}\cap\mathcal{A}=\mathbb{C}$
and
$\mathcal{B}{{}^{\prime}}\cap\mathcal{D}\subseteq\mathcal{B}{{}^{\prime}}\cap\mathcal{A}=\mathbb{C}$;
so, by [24, Corollary 1.4.3] again, $F_{u}$ and
$E_{\restriction_{\mathcal{D}}}$ are both unique and, hence, minimal. Thus,
from [16, Theorem 3], we get
$[\mathcal{A}:\mathcal{B}]_{0}=[\mathcal{A}:\mathcal{D}]_{0}[\mathcal{D}:\mathcal{B}]_{0}$
for every $\mathcal{D}\in\mathrm{Orb}_{G}(\mathcal{C})$. Note that,
$[\mathcal{A}:\mathcal{D}]_{0}=[\mathcal{A}:u\mathcal{C}u^{*}]_{0}=[\mathcal{A}:\mathcal{C}]_{0}$,
by [9, Lemma 2.8], which then shows that
$[\mathcal{D}:\mathcal{B}]_{0}=[\mathcal{C}:\mathcal{B}]_{0}$, as well. ∎

##  5\. Comparisons between various notions of distance

In this section, we discuss the various notions of distance between
subalgebras of a given operator algebra by Christensen and Mashood-Taylor and
make comparisons between them and the Kadison-Kastler distance.

###  5.1. Christensen’s distances

Christensen has given two notions of distances and used them effectively in
proving some significant perturbation results. The first one is defined
between subalgebras of any tracial von Neumann algebra ([5]) and, more
generally, the other one is defined between subalgebras of any $C^{*}$-algebra
([6]). We briefly recall both notions in reversed order and state some facts
related to them.

####  5.1.1. Christensen’s distance between subalgebras of a normed algebra

Though Christensen gave the notion of distance between subalgebras of a
$C^{*}$-algebra, the same can be used in the normed algebra context as well.

Let $A$ be a normed algebra. Recall from [6] that, for
$C,\,D\in\mathrm{Sub}_{A}$ and a scalar $\gamma>0$, $C\subseteq_{\gamma}D$ if
for each $x\in B_{1}(C)$, there exists a $y\in D$ such that
$\|x-y\|\leq\gamma$; and, the Christensen distance between $C$ and $D$ is
defined by

(5.1) $d_{0}(C,D)=\inf\\{\gamma>0\,:\,C\subseteq_{\gamma}D\text{ and
}D\subseteq_{\gamma}C\\}.$

Here are some elementary observations regarding the Christensen distance, some
of which will be used ahead.

######  Remark 5.1.

Let $A$ be a normed algebra.

  1. (1)

$d_{0}(C,D)\leq 1$ for all $C,D\in\text{Sub}_{A}$.

  2. (2)

$d_{0}(C,D)=d_{0}(\overline{C},D)=d_{0}(\overline{C},\overline{D})$ for all
$C,D\in\text{Sub}_{A}$.

  3. (3)

$d_{0}$ is not a metric on $\mathrm{Sub}_{A}$ (as it does not satisfy the
triangle inequality). However, $d_{0}$ and $d_{KK}$ are “equivalent” in the
sense that

$d_{0}(C,D)\leq d_{KK}(C,D)\leq 2d_{0}(C,D)$

for all $C,D\in\mathrm{Sub}_{A}$. ([7, Remark 2.3])

  4. (4)

If $C,D\in\mathrm{Sub}_{A}$ and $C$ is a norm closed proper subalgebra of $D$,
then $d_{0}(C,D)=1$. (Follows from [7, Proposition 2.4] \- also compare with
Remark 3.2(3).)

####  5.1.2. Christensen’s distance between subalgebras of a tracial von
Neumann algebra

For a von Neumann algebra $\mathcal{M}$, let

$*\text{-Sub}^{u}_{\mathcal{M}}:=\\{\text{ *-subalgebras of }\mathcal{M}\text{
containing }1_{\mathcal{M}}\\}\text{ and }$
$W^{*}\text{-}\mathrm{Sub}_{\mathcal{M}}:=\\{\text{von Neumann subalgebras of
}\mathcal{M}\ \text{(\,possibly with different unit)}\\}.$

Let $\mathcal{M}$ be a von Neumann algebra with a (fixed) faithful normal
tracial state $\tau$. Then, $\mathcal{M}$ inherits a natural inner product
structure with respect to the inner product $\langle
x,y\rangle_{\tau}=\tau(y^{*}x)$, $x,y\in\mathcal{M}$. The corresponding norm
on $\mathcal{M}$ will be denoted by $\|x\|_{\tau}$, i.e.,
$\|x\|_{\tau}=\tau(x^{*}x)^{1/2}$, $x\in\mathcal{M}$; and, the Hilbert space
completion of $\mathcal{M}$ is denoted by $L^{2}(\mathcal{M},\tau)$. There is
a natural embedding of $\mathcal{M}$ into $B(L^{2}(\mathcal{M},\tau))$ via
left multiplication and this allows us to consider $\mathcal{M}$ as a von
Neumann algebra on $L^{2}(\mathcal{M},\tau)$.

Further, in order to distinguish the elements of the inner-product structure
$\mathcal{M}$ from that of the von Neumann algebra $\mathcal{M}$, as is
standard, we shall use the notation $\widehat{x}$ for the elements of the
inner-product space $\mathcal{M}$. Thus,
$\langle\widehat{x},\widehat{y}\rangle_{\tau}=\tau(y^{*}x)$ and
$x(\widehat{1})=\widehat{x}$ for all $x,y\in\mathcal{M}$.

Recall from [5] that, for $P,\,Q\in\mathrm{Sub}_{\mathcal{M}}$ and a scalar
$\gamma>0$, $P\subset_{\gamma}Q$ if for each $x\in B_{1}(P)$ (w.r.t.
$\|\cdot\|$), there exists a $y\in Q$ such that
$\|\widehat{x}-\widehat{y}\|_{\tau}\leq\gamma$; and, the Christensen distance
between $P$ and $Q$ is defined by

(5.2)
$d_{C}(P,Q)=\inf\\{\gamma>0\,|\,P\subset_{\gamma}Q\,\text{and}\,Q\subset_{\gamma}P\\}.$

Here are some facts related to the Christensen distance $d_{C}$.

######  Remark 5.2.

With running notations, the following hold:

  1. (1)

$d_{C}(P,Q)\leq 1$ for all $P,Q\in\mathrm{Sub}_{\mathcal{M}}$.

  2. (2)

$d_{C}$ is a complete metric on $W^{*}\text{-}\mathrm{Sub}_{\mathcal{M}}\,$.
([5, Theorem, 5.1])

  3. (3)

From Proposition 5.8 and Proposition 5.5 ahead, it follows that

$d_{C}(P,Q)=d_{C}(\overline{P}^{S.O.T.},\overline{Q}^{S.O.T.})=d_{C}(P^{\prime\prime},Q^{\prime\prime})$

for all $P,Q\in*\text{-}\mathrm{Sub}_{\mathcal{M}}^{u}$.

Here is a well known observation which will be essential for our discussion -
see, for instance, [1, Proposition 2.6.4].

######  Proposition 5.3.

Let $M$ be a unital $C^{*}$-algebra equipped with a faithful tracial state
$\tau$. Then, $M$ is a von Neumann algbra on $L^{2}(M,\tau)$ iff
$\widehat{B_{1}(M)}$ is complete w.r.t. $\|.\|_{\tau}$.

In particular, $M$ is a $W^{*}$-algebra iff $\widehat{B_{1}(M)}$ is complete
w.r.t. $\|.\|_{\tau}$.

###  5.2. Mashood- Taylor distance between subalgebras of a tracial von
Neumann algebra

Given a $II_{1}$-factor $M$ with a unique faithful normal tracial state $\tau$
and any two subfactors $P$ and $Q$ of $M$, Mashood and Taylor (in [20])
consider the Hausdorff distance (w.r.t. $\|\cdot\|_{\tau}$) between
${B_{1}(P)}$ and ${B_{1}(Q)}$ and prove some nice results related to
continuity of ‘Jones’ index. They also mention (on [20, Page 56]) that this
distance gives the same topology on the set of subfactors of $M$ as the one
given by the Christensen’s distance $d_{C}$. We shall show in this section
that $d_{C}=d_{MT}$.

Mashood-Taylor distance can, in fact, be defined in a slightly more general
setup as follows:

For a von Neumann algebra $\mathcal{M}$ with a faithful normal tracial state
$\tau$, for any pair $P,Q\in\text{Sub}_{\mathcal{M}}$, the Mashood-Taylor
distance between $P$ and $Q$ is defined as

(5.3)
$d_{MT}(P,Q)=d_{H,\|\cdot\|_{\tau}}(\widehat{B_{1}(P)},\widehat{B_{1}(Q)}),$

where $d_{H,\|\cdot\|_{\tau}}$ denotes the Hausdorff distance with respect to
the metric induced by the norm $\|\cdot\|_{\tau}$ and $\widehat{S}\subseteq
L^{2}(\mathcal{M},\tau)$ for $S\subseteq\mathcal{M}$ (as in Section 5.1.2).

######  Lemma 5.4.

Let $\mathcal{M}$ be a von Neumann algebra with a faithful normal tracial
state $\tau$. Then, the following hold:

  1. (1)

$d_{MT}$ is a semi-metric on $\mathrm{Sub}_{\mathcal{M}}$.

  2. (2)

$d_{MT}$ is a metric on $W^{*}\text{-}\mathrm{Sub}_{\mathcal{M}}$.

  3. (3)

$d_{C}(P,Q)\leq d_{MT}(P,Q)$ for all $P,Q\in\text{Sub}_{\mathcal{M}}$.

###### Proof.

(1) follows readily from the definition.

(2): Note that $\widehat{B_{1}(\mathcal{Q})}$ is closed and bounded in
$L^{2}(\mathcal{M},\tau)$ for every $\mathcal{Q}\in
W^{*}\text{-Sub}_{\mathcal{M}}$ by Proposition 5.3. And, it is well known that
the Hausdorff distance is a metric on the collection of closed and bounded
subsets of a metric space.

(3): Let $P,Q\in\text{Sub}_{\mathcal{M}}$ and $\epsilon>0$. By the definition
of $d_{MT}(P,Q)$, for each $x\in B_{1}(P)$, there exists a $z\in B_{1}(Q)$
such that $\|\widehat{x}-\widehat{z}\|_{\tau}\leq d_{MT}(P,Q)+\epsilon$. This
implies that $P\subset_{d_{MT}(P,Q)+\epsilon}Q$. Similiarly, we have
$Q\subset_{d_{MT}(P,Q)+\epsilon}P$. Thus, $d_{C}(P,Q)\leq
d_{MT}(P,Q)+\epsilon$ for all $\epsilon>0$ and we are done.  ∎

The following observation is a nice tool to calculate the distance between two
von Neumann subalgebras and will be used ahead on more than occasion.

######  Proposition 5.5.

Let $(\mathcal{M},\tau)$ be as in Lemma 5.4. Then,

$d_{MT}(P,Q)=d_{MT}\Big{(}P,\overline{Q}^{S.O.T.}\Big{)}=d_{MT}\Big{(}\overline{P}^{S.O.T.},\overline{Q}^{S.O.T.}\Big{)}=d_{MT}\Big{(}P^{\prime\prime},Q^{\prime\prime}\Big{)}$

for all $P,Q\in*\text{-}\mathrm{Sub}^{u}_{\mathcal{M}}$.

###### Proof.

In order to avoid any possible confusion, for every $x\in\mathcal{M}$ and
$S\subseteq\mathcal{M}$, let
$d_{\tau}\big{(}\widehat{x},\widehat{S}\big{)}:=\inf\\{\|\widehat{x}-\widehat{s}\|_{\tau}:s\in
S\\}$.

Now, let $P,Q\in*\text{-Sub}^{u}_{\mathcal{M}}$. We first assert that
$d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(Q)}\Big{)}=d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}$
for all $a\in B_{1}(P)$.

Clearly,
$d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.}\Big{)}}\leq
d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(Q)}\Big{)}$ for all $a\in B_{1}(P)$.
For the reverse inequality, fix an $a\in B_{1}(P)$ and an $\epsilon>0$. Then,
there exists a $b_{0}\in B_{1}(\overline{Q}^{S.O.T.})$ such that

$d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}\leq\|\widehat{a}-\widehat{b_{0}}\|_{\tau}<d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}+\epsilon.$

As $B_{1}\Big{(}\overline{Q}^{S.O.T.}\Big{)}=\overline{B_{1}(Q)}^{S.O.T.}$ (by
Kaplansky density theorem), there exists a net $\\{b_{\alpha}\\}\subseteq
B_{1}(Q)$ such that $b_{\alpha}\stackrel{{\scriptstyle
S.O.T.}}{{\longrightarrow}}b_{0}$. Hence, there exists an $\alpha_{0}$ such
that

$\|\widehat{a}-\widehat{b}_{\alpha_{0}}\|_{\tau}<d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}+\epsilon.$

This implies that

$d_{\tau}\big{(}\widehat{a},\widehat{B_{1}(Q)}\big{)}\leq\|\widehat{a}-\widehat{b}_{\alpha_{0}}\|_{\tau}<d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}+\epsilon.$

As $\epsilon>0$ was arbitrary, we get
$d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(Q)}\Big{)}\leq
d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}$.
This proves our assertion.

Then, we get

$\beta:=\sup_{a\in
B_{1}(P)}d_{\tau}\big{(}\widehat{a},\widehat{B_{1}(Q)}\big{)}=\sup_{a\in
B_{1}(P)}d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}$
$\leq\sup_{z\in
B_{1}(\overline{P}^{S.O.T.})}d_{\tau}\Big{(}\widehat{z},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}=:\alpha.$

Further, for any $\eta>0$, there exists a $z_{0}\in
B_{1}\Big{(}\overline{P}^{S.O.T.}\Big{)}$ such that

$\alpha-\eta<d_{\tau}\Big{(}\widehat{z_{0}},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}\leq\alpha.$

Then, by Kaplansky density theorem again, there exists a net
$\\{z_{\lambda}\\}\subset B_{1}(P)$ such that
$z_{\lambda}\stackrel{{\scriptstyle S.O.T.}}{{\longrightarrow}}z_{0}$. In
particular, $\widehat{z_{\lambda}}{\longrightarrow}\widehat{z_{0}}$ with
respect to $\|\cdot\|_{\tau}$; and, since $d_{\tau}$ is continuous with
respect to $\|\cdot\|_{\tau}$, it follows that
$d_{\tau}\Big{(}\widehat{{z}_{\lambda}},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}\rightarrow
d_{\tau}\Big{(}\widehat{z_{0}},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}.$
Thus, there exists a $\lambda_{0}$ such that
$d_{\tau}\Big{(}\widehat{z}_{\lambda_{0}},\widehat{B_{1}(\overline{Q}^{S.O.T.})}\Big{)}>\alpha-\eta;$
so that, $\beta\geq\alpha-\eta$. Since $\eta>0$ was arbitrary, we have
$\beta\geq\alpha$. Thus, $\alpha=\beta$, i.e.,
$d_{MT}(P,Q)=d_{MT}\Big{(}{P},\overline{Q}^{S.O.T.}\Big{)}.$ By a similar
argument, we conclude that

$d_{MT}(P,Q)=d_{MT}\Big{(}\overline{P}^{S.O.T.},\overline{Q}^{S.O.T.}\Big{)}.$

Rest follows from von Neumann’s Double Commutant theorem. ∎

###  5.3. Comparisons between $d_{KK}$, $d_{C}$ and $d_{MT}$

We first present the following comparison between $d_{MT}$ and $d_{KK}$.

######  Lemma 5.6.

Let $\mathcal{M}$ be a von Neumann algebra with a faithful normal tracial
state $\tau$. Then,

$d_{MT}(P,Q)\leq d_{KK}(P,Q)$

for all $P,Q\in\mathrm{Sub}_{\mathcal{M}}$.

###### Proof.

Let $P,Q\in\mathrm{Sub}_{\mathcal{M}}$ and $p\in B_{1}(P)$. Then,

$d_{\tau}\Big{(}\widehat{p},\widehat{B_{1}(Q)}\Big{)}\leq\|\widehat{p}-\widehat{q}\|_{\tau}\leq\|p-q\|$

for all $q\in B_{1}(Q)$. Thus,
$d_{\tau}\Big{(}\widehat{p},\widehat{B_{1}(Q)}\Big{)}\leq
d\big{(}p,B_{1}(Q)\big{)}$ for all $p\in B_{1}(P)$. In particular,

$\sup_{p\in
B_{1}(P)}d_{\tau}\Big{(}\widehat{p},\widehat{B_{1}(Q)}\Big{)}\leq\sup_{p\in
B_{1}(P)}d\big{(}p,B_{1}(Q)\big{)}.$

Similarly,

$\sup_{q\in
B_{1}(Q)}d_{\tau}\Big{(}\widehat{q},\widehat{B_{1}(P)}\Big{)}\leq\sup_{q\in
B_{1}(Q))}d\big{(}q,B_{1}(P)\big{)}.$

Hence,

$d_{MT}(P,Q)\leq d_{KK}(P,Q).$

∎

The following useful observation is well known to the experts (see, for
instance, [5, Eqn.(6)]).

######  Lemma 5.7.

Let $(\mathcal{M},\tau)$ be as in Lemma 5.6 and let $\mathcal{N}$ be a von
Neumann subalgebra of $\mathcal{M}$ (with common unit). Then,

$d_{\tau}\Big{(}\widehat{x},\widehat{B_{1}(\mathcal{N})}\Big{)}=\|\widehat{x}-\widehat{E_{\mathcal{N}}(x)}\|_{\tau}=d_{\tau}\Big{(}\widehat{x},L^{2}(\mathcal{N})\Big{)}$

for all $x\in B_{1}(\mathcal{M})$, where $E_{\mathcal{N}}$ denotes the unique
$\tau$-preserving normal conditional expectation from $\mathcal{M}$ onto
$\mathcal{N}$.

We are now all set to show that the Mashood-Taylor distance agrees with the
Christensen’s distance on $*\text{-Sub}_{M}^{u}$.

######  Proposition 5.8.

Let $(\mathcal{M},\tau)$ be as in Lemma 5.7. Then,

$d_{C}(P,Q)=d_{MT}(P,Q)$

for all $P,Q\in*\text{-}\mathrm{Sub}_{\mathcal{M}}^{u}$.

###### Proof.

Let $P,Q\in*\text{-}\mathrm{Sub}_{\mathcal{M}}^{u}$ and consider
$\widetilde{P}:=\overline{P}^{S.O.T.}$ and
$\widetilde{Q}:=\overline{Q}^{S.O.T.}$. In view of Lemma 5.4 and Proposition
5.5, it suffices to show that $d_{C}(P,Q)\geq
d_{MT}\big{(}\widetilde{P},\widetilde{Q}\big{)}$.

We first assert that
$d_{\tau}\Big{(}\widehat{p},\widehat{(B_{1}(\widetilde{Q}))}\Big{)}=d_{\tau}\Big{(}\widehat{p},L^{2}(\widetilde{Q})\Big{)}=d_{\tau}\Big{(}\widehat{p},\widehat{Q}\Big{)}$
for all $p\in B_{1}(P)$.

Let $p\in B_{1}(P)$. Then, by Lemma 5.7,

$d_{\tau}\left(\widehat{p},\widehat{B_{1}(\widetilde{Q})}\right)=\|\widehat{p}-\widehat{E_{\widetilde{Q}}(p)}\|_{\tau}=d_{\tau}\left(\widehat{p},L^{2}(\widetilde{Q})\right).$

Thus, as $\widehat{B_{1}(Q)}\subset\widehat{Q}\subset L^{2}(\widetilde{Q})$,
we see that

$d_{\tau}\left(\widehat{p},\widehat{B_{1}(\widetilde{Q})}\right)=d_{\tau}\left(\widehat{p},L^{2}(\widetilde{Q})\right)\leq
d_{\tau}\left(\widehat{p},\widehat{Q}\right)\leq
d_{\tau}\left(\widehat{p},\widehat{B_{1}(Q)}\right)=d_{\tau}\left(\widehat{p},\widehat{B_{1}(\widetilde{Q})}\right)$

for all $p\in B_{1}(P)$. Hence, our first assertion is true.

Now, suppose that $P\subset_{\gamma}Q$ and $Q\subset_{\gamma}P$ for some
$\gamma>0$. We then assert that

$d_{\tau}\left(\widehat{p},\widehat{B_{1}(\widetilde{Q})}\right)\leq\gamma\text{
for all }p\in B_{1}(\widetilde{P}).$

So, let $p\in B_{1}(\widetilde{P})$ and, by Kaplansky density theorem, fix a
net $\\{p_{\alpha}\\}\subset B_{1}(P)$ such that
$p_{\alpha}\stackrel{{\scriptstyle S.O.T.}}{{\longrightarrow}}p$ in
$B(L^{2}(\mathcal{M},\tau))$. Since $P\subset_{\gamma}Q$, for each $\alpha$,
there exists a $q_{\alpha}\in Q$ such that
$\|\widehat{p}_{\alpha}-\widehat{q}_{\alpha}\|_{\tau}\leq\gamma.$ So,

$\gamma\geq\|\widehat{p}_{\alpha}-\widehat{q}_{\alpha}\|_{\tau}\geq
d_{\tau}(\widehat{p}_{\alpha},\widehat{Q})=d_{\tau}\left(\widehat{p}_{\alpha},\widehat{B_{1}(\widetilde{Q})}\right)=\|\widehat{p}_{\alpha}-\widehat{E_{\widetilde{Q}}(p_{\alpha})}\|_{\tau}$

for all $\alpha$, where $E_{\widetilde{Q}}$ is the unique $\tau$-preserving
normal conditional expectation from $\mathcal{M}$ onto $\widetilde{Q}$. Thus,

$d_{\tau}\left(\widehat{p},\widehat{B_{1}(\widetilde{Q})}\right)\leq\inf_{\alpha}\|\widehat{p}-\widehat{E_{\widetilde{Q}}(p_{\alpha})}\|_{\tau}\leq\inf_{\alpha}\|\widehat{p}-\widehat{p}_{\alpha}\|_{\tau}+\inf_{\alpha}\|\widehat{p}_{\alpha}-\widehat{E_{\widetilde{Q}}(p_{\alpha})}\|_{\tau}\leq\gamma,$

where the last inequality holds because
$\|\widehat{p}_{\alpha}-\widehat{p}\|_{\tau}\rightarrow 0\,$ (as
$p_{\alpha}\rightarrow p$ in S.O.T.) and
$\|\widehat{p_{\alpha}}-\widehat{E_{\widetilde{Q}}(p_{\alpha})}\|_{\tau}\leq\gamma$
for all $\alpha$. Thus, our second assertion is also true.

As a consequence, $\sup_{p\in
B_{1}(\widetilde{P})}d_{\tau}\left(\widehat{p},\widehat{B_{1}(\widetilde{Q})}\right)\leq\gamma.$
Likewise, $\sup_{q\in
B_{1}(\widetilde{Q})}d_{\tau}\left(\widehat{q},\widehat{B_{1}(\widetilde{P})}\right)\leq\gamma$
as well. Hence, $d_{MT}(\widetilde{P},\widetilde{Q})\leq\gamma$, which then
implies that

$d_{MT}\big{(}\widetilde{P},\widetilde{Q}\big{)}\leq d_{C}(P,Q),$

and we are done. ∎

In view of Lemma 5.6, we also have the following:

######  Corollary 5.9.

Let $(\mathcal{M},\tau)$ be as in Lemma 5.7. Then,

$d_{C}(P,Q)\leq d_{KK}(P,Q)$

for all $P,Q\in\mathrm{Sub}_{\mathcal{M}}$.

##  6\. Kadison-Kastler and Christensen distance between subalgebras
corresponding to subgroups

###  6.1. Distances between crossed-product subalgebras associated to
subgroups

Let $G$ be a discrete group acting on a $C^{*}$-algebra $\mathcal{A}$ via the
map $\alpha:G\rightarrow\mathrm{Aut}(\mathcal{A})$. Consider the space
$C_{c}(G,\mathcal{A})$ consisting of compactly supported $\mathcal{A}$-valued
functions on $G$, which can be identified naturally with the vector space
$\\{\sum_{\text{finite}}a_{g}g\,\,:\,\,a_{g}\in\mathcal{A}\,,\,g\in G\\}$
consisting of formal finite sums. Further, $C_{c}(G,\mathcal{A})$ is a
$*$-algebra with respect to the (so called, twisted) multiplication given by
the convolution operation

$\Big{(}\sum_{s\in I}a_{s}s\Big{)}\Big{(}\sum_{t\in J}b_{t}t\Big{)}=\sum_{s\in
I,t\in J}a_{s}\alpha_{s}(b_{t})st$

and involution given by

$\Big{(}\sum_{s\in I}a_{s}s\Big{)}^{*}=\sum_{s\in
I}\alpha_{s^{-1}}(a_{s}^{*})s^{-1}$

for any two finite sets $I$ and $J$ in $G$. The reduced crossed product
$\mathcal{A}\rtimes^{r}_{\alpha}G$ and the universal crossed product
$\mathcal{A}\rtimes^{u}_{\alpha}G$ are defined, respectively, as the
completions of $C_{c}(G,\mathcal{A})$ with respect to the reduced norm and the
universal norm on $C_{c}(G,\mathcal{A})$, as described below:

The reduced norm is given by

$\Big{\|}\sum_{\text{finite}}a_{g}g\Big{\|}_{r}:=\Big{\|}\sum_{\text{finite}}\pi(a_{g})(1\otimes\lambda_{g})\Big{\|}_{B(H\otimes
l^{2}(G))},$

where $\mathcal{A}\subset B(H)$ is a (equivalently, any) fixed faithful
representation of $\mathcal{A}$, $\lambda:G\rightarrow B(l^{2}(G))$ is the
left regular representation and $\pi:\mathcal{A}\rightarrow B(H\otimes
l^{2}(G))$ is the representation satisfying
$\pi(a)(\xi\otimes\delta_{g})=\alpha_{g^{-1}}(a)(\xi)\otimes\delta_{g}$ for
all $\xi\in H$ and $g\in G$.

The universal norm is given by

$\|x\|_{u}:=\sup_{\pi}\|\pi(x)\|_{B(H_{\pi})}\,\mbox{\,\,for\,\,}\,x\in
C_{c}(G,\mathcal{A}),$

where the supremum runs over all (cyclic) $*$-homomorphisms
$\pi:C_{c}(G,\mathcal{A})\rightarrow B(H_{\pi})$. Note that
$\|x\|_{r}\leq\|x\|_{u}\ \text{ for all}\ x\in C_{c}(G,\mathcal{A}).$

We refer the reader to [4] for more on crossed products.

####  6.1.1. Distances between subalgebras of reduced crossed product

First, we gather some relevant facts related to the reduced crossed product
construction.

######  Remark 6.1.

Let $G$ and $\mathcal{A}$ be as above and $H$ be a subgroup of $G$.

  1. (1)

The canonical injective $*$-homomorphism

$C_{c}(H,\mathcal{A})\ni\sum_{\mathrm{finite}}a_{h}h\mapsto\sum_{\mathrm{finite}}a_{h}h\in
C_{c}(G,\mathcal{A})$

extends to an injective $*$-homomorphism from
$\mathcal{A}\rtimes^{r}_{\alpha}H$ into $\mathcal{A}\rtimes^{r}_{\alpha}G$.
Hence, we can consider $\mathcal{A}\rtimes^{r}_{\alpha}H$ as
$C^{*}$-subalgebra of $\mathcal{A}\rtimes^{r}_{\alpha}G$. ([17, Remark 3.2])

  2. (2)

There exists a faithful conditional expectation
$E_{H}:\mathcal{A}\rtimes_{\alpha}^{r}G\rightarrow\mathcal{A}\rtimes_{\alpha}^{r}H$
satisfying $E_{H}(\sum_{g\in I}a_{g}g)=\sum_{h\in I\cap H}a_{h}h$ for all
$\sum_{g\in I}a_{g}g\in C_{c}(G,\mathcal{A})$ ([17, Remark 3.2]); and, if
$[G:H]<\infty$, then $E_{H}$ has finite (Watatani) index with a quasi-basis
given by any set of left coset representatives $\\{g_{i}:1\leq i\leq[G:H]\\}$
of $H$ in $G$.

  3. (3)

Thus, as in Section 2, the vector space $\mathcal{A}\rtimes^{r}_{\alpha}G$ is
a natural left $(\mathcal{A}\rtimes^{r}_{\alpha}H)$-module via multiplication
from left. Hence, following [24], we consider
$\mathcal{A}\rtimes^{r}_{\alpha}G$ as a pre-Hilbert (left)
$(\mathcal{A}\rtimes^{r}_{\alpha}H)$-module with respect to the
$(\mathcal{A}\rtimes^{r}_{\alpha}H)$-valued inner product given by

$\langle x,y\rangle=E_{H}(x^{*}y),\text{ for
}x,y\in\mathcal{A}\rtimes^{r}_{\alpha}G.$

Further, in order to distinguish the elements of the $C^{*}$-algebra
$\mathcal{A}\rtimes^{r}_{\alpha}G$ and the pre-Hilbert
$(\mathcal{A}\rtimes^{r}_{\alpha}H)$-module
$\mathcal{A}\rtimes^{r}_{\alpha}G$, following [24], we consider the identity
map
$\eta_{H}:\mathcal{A}\rtimes^{r}_{\alpha}G\rightarrow\mathcal{A}\rtimes^{r}_{\alpha}G$,
where we consider the co-domain as the pre-Hilbert
$(\mathcal{A}\rtimes^{r}_{\alpha}H)$-module. Since $E_{H}$ is a contraction,
we have

(6.1) $\|\eta_{H}(x)\|=\|E_{H}(x^{*}x)\|^{1/2}_{r}\leq\|x\|_{r}$

for all $x\in\mathcal{A}\rtimes^{r}_{\alpha}G$.

  4. (4)

Note that, for two subgroups $H\subsetneq K$ of a discrete group $G$,
$\|\eta_{H}(x)\|$ need not be equal to $\|\eta_{K}(x)\|$ for every
$x\in\mathcal{A}\rtimes^{r}_{\alpha}G$.

For instance, consider the trivial action of the finite permutation group
$G=S_{3}$ on the $C^{*}$-algebra $\mathbb{C}$. Let $H=\\{e\\}$, $K=A_{3}$ and
$x=a_{1}(123)+a_{2}(132)\in C_{r}^{*}(G)=\mathbb{C}[G]$ with $a_{1},a_{2}\neq
0$. Then,
$x^{*}x=(|a_{1}|^{2}+|a_{2}|^{2})e+\bar{a_{1}}a_{2}(123)+\bar{a_{2}}a_{1}(132)$
and

$\displaystyle\|\eta_{K}(x)\|^{2}$ $\displaystyle=$
$\displaystyle\|E_{K}(x^{*}x)\|_{r}$ $\displaystyle=$
$\displaystyle\Big{\|}(|a_{1}|^{2}+|a_{2}|^{2}){e}+(\bar{a_{1}}a_{2})\,{(123)}+(\bar{a_{2}}a_{1})\,{(132)}\Big{\|}_{r}$
$\displaystyle=$
$\displaystyle\Big{\|}(|a_{1}|^{2}+|a_{2}|^{2})\lambda_{e}+(\bar{a_{1}}a_{2})\,\lambda_{(123)}+(\bar{a_{2}}a_{1})\,\lambda_{(132)}\Big{\|}_{B(\ell^{2}(S_{3}))}$
$\displaystyle\geq$
$\displaystyle\Big{\|}\Big{(}(|a_{1}|^{2}+|a_{2}|^{2})\lambda_{e}+(\bar{a_{1}}a_{2})\,\lambda_{(123)}+(\bar{a_{2}}a_{1})\,\lambda_{(132)}\Big{)}(\delta_{e})\Big{\|}_{\ell^{2}(S_{3})}$
$\displaystyle=$
$\displaystyle\Big{\|}(|a_{1}|^{2}+|a_{2}|^{2})\delta_{e}+(\bar{a_{1}}a_{2})\,\delta_{(123)}+(\bar{a_{2}}a_{1})\,\delta_{(132)}\Big{\|}_{\ell^{2}(S_{3})}$
$\displaystyle=$
$\displaystyle\Big{(}(|a_{1}|^{2}+|a_{2}|^{2})^{2}+|\bar{a_{1}}a_{2}|^{2}+|\bar{a_{2}}a_{1}|^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle>$
$\displaystyle\Big{(}(|a_{1}|^{2}+|a_{2}|^{2})^{2}\Big{)}^{\frac{1}{2}}$
$\displaystyle=$ $\displaystyle(|a_{1}|^{2}+|a_{2}|^{2})$ $\displaystyle=$
$\displaystyle\|\eta_{H}(x)\|^{2}.$

  5. (5)

We shall denote $E_{\\{e\\}}$ (resp., $\eta_{\\{e\\}}$) simply by $E$ (resp.,
$\eta$).

The following is an elementary and useful observation. A more general version
of it was proved by Phillips (see [21, Proposition 9.16(3)]).

######  Lemma 6.2.

Let $G$ be a discrete group acting on a $C^{*}$-algebra $\mathcal{A}$. If
$a=\sum_{g\in I}a_{g}g\in C_{c}(G,\mathcal{A})$, then

$\|a_{g}\|^{2}\leq\|E(a^{*}a)\|_{r}=\|\eta(a)\|^{2}\text{ for all }\,g\in I.$

######  Proposition 6.3.

Let $G$, $\mathcal{A}$ be as in Lemma 6.2 and let $H$ and $K$ be two distinct
subgroups of $G$. Then,

$d_{KK}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))=1=d_{0}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))$

in $\mathcal{A}\rtimes^{r}_{\alpha}G$.

###### Proof.

Note that, $d_{KK}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))\leq 1$, by
definition and $d_{0}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))\leq
d_{KK}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))$, by Remark 5.1. So, it just
remains to show that

$d_{0}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))\geq 1.$

Since $H$ and $K$ are distinct, either $H\neq H\cap K$ or $K\neq H\cap K$.
Without loss of generality, we can assume that $H\neq H\cap K$. Then, in view
of (6.1) and Lemma 6.2, we observe that

$\|h-x\|_{r}\geq\|\eta(h-x)\|\geq 1>\gamma$

for all $h\in H\setminus H\cap K$, $x\in B_{1}(C_{c}(K,\mathcal{A}))$ and for
every $0<\gamma<1$. Thus, if
$C_{c}(H,\mathcal{A})\subseteq_{\gamma}C_{c}(K,\mathcal{A})$ for some
$\gamma>0$, then $\gamma\geq 1$. So, by the definition of $d_{0}$, we must
have

$d_{0}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))\geq 1.$

∎

####  6.1.2. Distances between subalgebras of universal crossed product

######  Remark 6.4.

Let $G$, $\mathcal{A}$ and $\alpha:G\rightarrow\mathrm{Aut}(\mathcal{A})$ be
as in the preceding subsection and $H$ be a subgroup of $G$. Then, the
canonical injective $*$-homomorphism

$C_{c}(H,\mathcal{A})\ni\sum_{\mathrm{finite}}a_{h}h\mapsto\sum_{\mathrm{finite}}a_{h}h\in
C_{c}(G,\mathcal{A})$

extends to an injective $*$-homomorphism from
$\mathcal{A}\rtimes^{u}_{\alpha}H$ into $\mathcal{A}\rtimes^{u}_{\alpha}G$.
Hence, we can consider $\mathcal{A}\rtimes^{u}_{\alpha}H$ as
$C^{*}$-subalgebra of $\mathcal{A}\rtimes^{u}_{\alpha}G$. ([17, Proposition
3.1])

Since $\|x\|_{u}\geq\|x\|_{r}$ on $C_{c}(G,\mathcal{A})$, the following is
immediate from the proof of Proposition 6.3.

######  Proposition 6.5.

Let $G$, $\mathcal{A}$ be as in Lemma 6.2 and let $H$ and $K$ be two distinct
subgroups of $G$. Then,

$d_{KK}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))=1=d_{0}(C_{c}(H,\mathcal{A}),C_{c}(K,\mathcal{A}))$

in $\mathcal{A}\rtimes^{u}_{\alpha}G$.

In view of Lemma 3.3, its analogue in Remark 5.1 and the preceding two
propositions, we obtain:

######  Corollary 6.6.

Let $G,H,K$, $\mathcal{A}$ and $\alpha$ be as in Proposition 6.3. Then,

  1. (1)

$d_{KK}(A\rtimes^{r}_{\alpha}H,A\rtimes^{r}_{\alpha}K)=1=d_{0}(A\rtimes^{r}_{\alpha}H,A\rtimes^{r}_{\alpha}K)$
in $\mathcal{A}\rtimes^{r}G$; and,

  2. (2)

$d_{KK}(A\rtimes^{u}_{\alpha}H,A\rtimes^{u}_{\alpha}K)=1=d_{0}(A\rtimes^{u}_{\alpha}H,A\rtimes^{u}_{\alpha}K)$
in $A\rtimes^{u}_{\alpha}G$.

####  6.1.3. Some observations related to the $C^{*}$-algebras associated to
groups

Recall that for any discrete group $G$, if $\mathcal{A}=\mathbb{C}$ and
$\alpha:G\to\mathrm{Aut}(G)$ is the trivial action, then
$\mathcal{A}\rtimes^{r}_{\alpha}G$ (resp., $\mathcal{A}\rtimes^{u}_{r}G$) is
just the reduced group $C^{*}$-algebra $C^{*}_{r}(G)$ (resp., the (universal)
group $C^{*}$-algebra $C^{*}_{u}(G)$). Thus, we readily obtain the following:

######  Corollary 6.7.

Let $G$ be a discrete group and $H$ and $K$ be two distinct subgroups of $G$.
Then,

  1. (1)

$d_{KK}(\mathbb{C}[H],\mathbb{C}[K])=d_{KK}(C^{*}_{r}(H),C^{*}_{r}(K))=1=d_{0}(\mathbb{C}[H],\mathbb{C}[K])=d_{0}(C^{*}_{r}(H),C^{*}_{r}(K))$
in $C^{*}_{r}(G)$; and,

  2. (2)

$d_{KK}(\mathbb{C}[H],\mathbb{C}[K])=d_{KK}(C^{*}_{u}(H),C^{*}_{u}(K))=1=d_{0}(\mathbb{C}[H],\mathbb{C}[K])=d_{0}(C^{*}_{u}(H),C^{*}_{u}(K))$
in $C^{*}_{u}(G)$.

As in Remark 6.1, for any subgroup $H$ of $G$, we consider the identity map
$\eta_{H}:C^{*}_{r}(G)\to C^{*}_{r}(G)$ with the pre-Hilbert
$C^{*}_{r}(H)$-norm $\|\eta_{H}(x)\|:=\|E_{H}(x^{*}x)\|_{r}^{1/2}$ for all
$x\in\mathbb{C}[G]$. Also, we simply write $\eta$ for $\eta_{e}$.

######  Lemma 6.8.

With running notations,

(6.2) $\left\|\eta\left(\sum_{g\in
G}\alpha_{g}g\right)\right\|^{2}\leq\sum_{g\in G}|\alpha_{g}|\left(\sum_{x\in
gH}|\alpha_{x}|\right)$

for all $\sum_{g}\alpha_{g}g\in\mathbb{C}[G]$.

###### Proof.

Note that

$E\left(\Big{(}\sum_{g\in G}\alpha_{g}g\Big{)}^{*}\Big{(}\sum_{g\in
G}\alpha_{g}g\Big{)}\right)=\sum_{g\in G}\sum_{x\in
gH}\bar{\alpha_{g}}\alpha_{x}g^{-1}x=\sum_{g}\bar{\alpha_{g}}g^{-1}\Big{(}\sum_{x\in
gH}\alpha_{x}x\Big{)}.$

for all $\sum_{g}\alpha_{g}g\in\mathbb{C}[G]$. ∎

Further, the reduced group $C^{*}$-algebra always admits a canonical faithful
tracial state $\tau:C^{*}_{r}(G)\to\mathbb{C}$ which satisfies
$\tau(x)=\langle\lambda(x)\delta_{e},\delta_{e}\rangle$ for all
$x\in\mathbb{C}[G]$, where $\lambda:\mathbb{C}[G]\to B(\ell^{2}(G))$ is the
faithful $*$-representation induced by the (left) regular representation of
$\mathbb{C}[G]$.

######  Remark 6.9.

Let $G$ be a discrete group.

  1. (1)

If $H$ is the trivial subgroup of $G$, then the conditional expectation $E$
from $C^{*}_{r}(G)$ onto $\mathbb{C}\ (\equiv C^{*}_{r}(H))$ is just the
tracial state $\tau$ on $C^{*}_{r}(G)$, which satisfies $E(\sum_{g\in
G}\alpha_{g}g)=\alpha_{e}$ for all $\sum_{g}\alpha_{g}g\in\mathbb{C}[G]$, the
$\mathbb{C}$-valued inner product induced by $E$ is just the usual inner
product on $C^{*}_{r}(G)$ induced by $\tau$, and

(6.3) $\left\|\eta(\sum_{g\in G}\alpha_{g}g)\right\|=\sqrt{\sum_{g\in
G}\left|\alpha_{g}\right|^{2}}$

for all $\sum_{g}\alpha_{g}g\in\mathbb{C}[G]$ because

$E\left((\sum_{g\in G}\alpha_{g}g)^{*}(\sum_{g\in
G}\alpha_{g}g)\right)=\sum_{g\in G}\left|\alpha_{g}\right|^{2}.$

  2. (2)

We thus deduce that
$\|\sum_{g}\alpha_{g}g\|_{r}\geq\sqrt{\Big{(}\sum_{g}|\alpha_{g}|^{2})}$ for
all $\sum_{g}\alpha_{g}g\in\mathbb{C}[G]$.

In particular, $\sqrt{2}\leq\|g-h\|_{r}\leq{2}$ for any two distinct elements
$g,h$ of $G$. Thus, $\\{g:g\in G\\}$ is a discrete linearly independent subset
of the unit sphere of $C^{*}_{r}(G)$.

  3. (3)

If $G$ is finite, then $C^{*}_{r}(G)=\mathbb{C}[G]$ and

(6.4) $\|\eta(x)\|\leq\|x\|_{r}\leq|G|\,\|\eta(x)\|$

for all  $x\in\mathbb{C}[G]$, because, for every
$x=\sum_{g}\alpha_{g}g\in\mathbb{C}[G]$, $\|\eta(x)\|=\sqrt{\sum_{g\in
G}\left|\alpha_{g}\right|^{2}}$ (and the last inequality in (6.4) follows from
the Hölder’s inequality).

Moreover, $E:\mathbb{C}[G]\to\mathbb{C}$ has finite index with a quasi-basis
$\\{g:g\in G\\}$ and $\mathrm{Ind}_{W}(E)=|G|$.

######  Remark 6.10.

If $\mathcal{B}$ is a $*$-subalgebra of a unital $C^{*}$-algebra
$\mathcal{A}$, then $\mathcal{N}_{\mathcal{A}}(\mathcal{B})$ is a subgroup of
$\mathcal{N}_{\mathcal{A}}(\overline{\mathcal{B}})$. In particular, for any
subgroup $H$ of a discrete group $G$,

$\mathcal{N}_{G}(H)\leq\mathcal{N}_{C^{*}_{r}(G)}(\mathbb{C}[H])\leq\mathcal{N}_{C^{*}_{r}(G)}(C^{*}_{r}(H)).$

Given a subgroup $H$ of a discrete group $G$ and a unitary $u$ in
$\mathcal{U}(C^{*}_{r}(G))\setminus\mathcal{N}_{\mathbb{C}[G]}(\mathbb{C}[H])$,
it is natural to ask whether $u\mathbb{C}[H]u^{*}=\mathbb{C}[K]$ or
$uC^{*}_{r}(H)u^{*}=C^{*}_{r}(K)$ for some subgroup $K$ of $G$ or not.
Obviously, for every $g\in G$, $g\mathbb{C}[H]g^{*}=\mathbb{C}[gHg^{-1}]$,
$gC^{*}_{r}(H)g^{*}=C^{*}_{r}(gHg^{-1})$ and
$\|g-\mathbf{1}\|_{r}\geq\sqrt{2}$. However, when $u\notin G$, then it is not
clear when $u\mathbb{C}[H]u^{*}$ equals $\mathbb{C}[K]$ for some subgroup $K$
of $G$. 6.7 allows us to deduce the following partial answer to this question.

######  Proposition 6.11.

Let $H$ be a proper subgroup of a discrete group $G$ and $u$ be a unitary in
$\mathbb{C}[G]$ (resp., $C^{*}_{r}(G)$) such that $\|u-\mathbf{1}\|_{r}<1/2$.
If $u\mathbb{C}[H]u^{*}=\mathbb{C}[K]$ (resp.,
$uC^{*}_{r}(H)u^{*}=C^{*}_{r}(K)$) for some subgroup $K$ of $G$, then $K=H$
and, in particular, $u\in\mathcal{N}_{\mathbb{C}[G]}(\mathbb{C}[H])$ (resp.,
$u\in\mathcal{N}_{C^{*}_{r}(G))}(C^{*}_{r}(H))$),

###### Proof.

We first discuss the group algebra case. Let $u\in\mathbb{C}[G]$ with
$\|u-\mathbf{1}\|_{r}<1/2$.

Suppose that $u\mathbb{C}[H]u^{*}=\mathbb{C}[K]$ for some subgroup $K$ of $G$.
Then, by Lemma 3.5, we observe that

$d_{KK}(\mathbb{C}[H],\mathbb{C}[K])=d_{KK}(\mathbb{C}[H],u\mathbb{C}[H]u^{*})\leq
2\|u-\mathbf{1}\|_{r}<1.$

Thus, $K=H$, by 6.7.

Even in the reduced group $C^{*}$-algebra case, when $u$ is a unitary in
$C^{*}_{r}(G)$ satisfying the inequality $\|u-\mathbf{1}\|_{r}<1/2$ and that
$uC^{*}_{r}(H)u^{*}=C^{*}_{r}(K)$ for some subgroup $K$ of $G$, the same
argument shows that $d_{KK}(C^{*}_{r}(H),C^{*}_{r}(K))<1$; so that, $K=H$ (by
6.7 again). ∎

Here are two obvious reformulations of the preceding corollary.

######  Remark 6.12.

Let $H$, $G$ be as in Proposition 6.11 and $u$ be a unitary in $\mathbb{C}[G]$
(resp., $C^{*}_{r}(G)$).

  1. (1)

If $u\mathbb{C}[H]u^{*}=\mathbb{C}[K]$ (resp.,
$uC^{*}_{r}(H)u^{*}=C^{*}_{r}(K)$) for some subgroup $K$ other than $H$, then
$\|u-\mathbf{1}\|_{r}\geq 1/2$.

  2. (2)

If $\|u-\mathbf{1}\|_{r}<1/2$, then the conjugate $*$-subalgebra
$u\mathbb{C}[H]u^{*}$ (resp., $uC^{*}_{r}(H)u^{*}$) is not equal to
$\mathbb{C}[K]$ (resp., $C^{*}_{r}(K)$) for any subgroup $K$ other than $H$.

######  Remark 6.13.

One could ask whether every unitary $u$ in
$\mathcal{N}_{\mathbb{C}[G]}(\mathbb{C}[H])$ (resp., in
$\mathcal{N}_{C^{*}_{r}(G)}(C^{*}_{r}(H))$) satisfies the inequality
$\|u-\mathbf{1}\|_{r}<1/2$. This is trivially seen to be false.

Indeed, if $H$ is a subgroup of $G$ with a non-trivial normalizer, then for
any $e\neq g\in\mathcal{N}_{G}(H)$,
$g\mathbb{C}[H]g^{*}=\mathbb{C}[gHg^{-1}]=\mathbb{C}[H]$ whereas, as noted in
Remark 6.9(2), we have

$\|g-\mathbf{1}\|_{r}=\|g-e\|_{r}\geq\sqrt{2}>1/2.$

By the same argument, one also concludes that if
$u\in\mathcal{N}_{C^{*}_{r}(G)}(C^{*}_{r}(H))$, then the inequality
$\|u-\mathbf{1}\|_{r}<1/2$ need not be true.

######  Remark 6.14.

Note that, by Lemma 3.4, we can always find a unitary $u$ in $C^{*}_{r}(G)$
such that $0<\|\mathbf{1}-u\|_{r}<\frac{1}{2}$. Hence, as is well known, not
every $C^{*}$-subalgebra of $C^{*}_{r}(G)$ is a reduced subgroup
$C^{*}$-algebra.

The following recipe provides a concrete way of obtaining unitaries
arbitrarily close to $\mathbf{1}$.

######  Lemma 6.15.

Let $G$ be a discrete group. If there exists an element $g\in G$ such that
$g=g^{-1}$, then $u_{\theta}:=\cos(\theta)e+i\sin(\theta)g$ is a unitary in
$\mathbb{C}[G]$ for every $\theta\in\mathbb{R}$. Also, for each $\epsilon>0$,
there exists a $\delta>0$ such that $\|u_{\theta}-\mathbf{1}\|_{r}<\epsilon$
whenever $|\theta|<\delta$.

###### Proof.

Let $\theta\in\mathbb{R}$. Then,

$\displaystyle u_{\theta}u_{\theta}^{*}$
$\displaystyle=(\cos(\theta)e+i\sin(\theta)g)(\cos(\theta)e-i\sin(\theta)g)$
$\displaystyle=(\cos^{2}(\theta)+\sin^{2}(\theta))e+(i\sin(\theta)\cos(\theta)-i\sin(\theta)\cos(\theta))g$
$\displaystyle=e.$

Similiarly, $u_{\theta}^{*}u_{\theta}=e$. Hence, $u_{\theta}$ is a unitary in
$\mathbb{C}[G]$ for every $\theta\in\mathbb{R}$.

Note that, for any given $\epsilon>0$, there exists a $\delta>0$ such that
$|\cos(\theta)-1|,|\sin(\theta)|<\epsilon/2$ whenever $|\theta|<\delta$. So,

$\|u_{\theta}-\mathbf{1}\|_{r}=\|(\cos(\theta)-1)e+i\sin(\theta)g\|_{r}\leq|\cos(\theta)-1|+|\sin(\theta)|<\epsilon$

whenever $|\theta|<\delta$. ∎

We can now find unitaries arbitrarily close to $\mathbf{1}$ yet not in the
normalizer of a subalgebra. Compare with 3.7.

######  Lemma 6.16.

Let $H$ be a proper subgroup of a discrete group $G$. If there exists an
element $g\in G\setminus H$ such that $g=g^{-1}$ and
$g\notin\mathcal{C}_{G}(H)$ (centralizer of $H$ in $G$), then
$u_{\theta}\notin\mathcal{N}_{C^{*}_{r}(G)}(C^{*}_{r}(H))$ for every
$\theta\in\mathbb{R}\setminus\\{\frac{n\pi}{2}:n\in\mathbb{Z}\\}$.

###### Proof.

Since $g\notin\mathcal{C}_{G}(H)$, there exists an $h\in H$ such that $gh\neq
hg$; also $gh,hg\notin H$. Then, for this $h\in\mathbb{C}[H]$ and
$\theta\in\mathbb{R}\setminus\\{\frac{n\pi}{2}:n\in\mathbb{Z}\\}$, we have

$u_{\theta}hu_{\theta}^{*}=\cos^{2}(\theta)h+i\sin(\theta)\cos(\theta)gh-i\sin(\theta)\cos(\theta)hg+\sin^{2}(\theta)ghg.$

Hence, $u_{\theta}\notin\mathcal{N}_{C^{*}_{r}(G)}(C^{*}_{r}(H))$ for every
$\theta\in\mathbb{R}\setminus\\{\frac{n\pi}{2}:n\in\mathbb{Z}\\}$. ∎

######  Example 6.17.

Let $G=S_{3}$ and $H=A_{3}$. Then,
$u_{\theta}:=\cos(\theta)e+i\sin(\theta)(12)$ is a unitary in
$\mathbb{C}[S_{3}]$ for every $\theta\in\mathbb{R}$. Clearly, by the preceding
lemma, $u_{\theta}\notin\mathcal{N}_{\mathbb{C}[S_{3}]}(\mathbb{C}[A_{3}])$
for every $\theta\in\mathbb{R}\setminus\\{\frac{n\pi}{2}:n\in\mathbb{Z}\\}$,
because $(12)\notin\mathcal{C}_{S_{3}}(A_{3})$. Also, we can choose
$0<\theta<\pi/2$ small enough so that $\|u_{\theta}-\mathbf{1}\|_{r}$ is as
small as we wish. Thus,

  1. (1)

for each $\epsilon>0$, there exists a unitary $u$ in $\mathbb{C}[S_{3}]$ such
that

$0<d_{KK}(\mathbb{C}[A_{3}],u\mathbb{C}[A_{3}]u^{*})<\epsilon;\text{and,}$

  2. (2)

$\mathbf{1}$ is not an interior point of
$\mathcal{N}_{\mathbb{C}[S_{3}]}(\mathbb{C}[A_{3}])$ in
$\mathcal{U}(\mathbb{C}[S_{3}])$.

###  6.2. Kadison-Kastler distance between (Banach) subgroup algebras

Recall that, for a discrete group $G$, the Banach space

$\ell^{1}(G):=\Big{\\{}f:G\rightarrow\mathbb{C}\,\,:\,\,\sum_{g\in
G}|f(g)|<\infty\Big{\\}}$

is a unital Banach $*$-algebra with multiplication given by convolution, i.e.,
for $a,b\in\ell^{1}(G),$

$(ab)(g):=\sum_{h\in G}a(h)b(h^{-1}g);$

and, involution given by

$a^{*}(g):=\overline{a(g^{-1})}$

for $a\in\ell^{1}(G),g\in G.$

Further, for each $g\in G,$ we define $u_{g}\in\ell^{1}(G)$ by
$u_{g}(h)=\delta_{g,h}$. Then, $u_{e}$ is the multiplicative identity for
$\ell^{1}(G)$.

######  Remark 6.18.

With running notations,

  1. (1)

There exists an injective unital ∗-homomorphism
$\mathit{i}:\mathbb{C}[G]\rightarrow\ell^{1}(G)$ such that $i(g)=u_{g}$ for
all $g\in G$ and the image of $i$ is dense in $\ell^{1}(G)$.

In particular, $\mathbb{C}[G]$ can be considered as a dense subspace of
$\ell^{1}(G)$.

  2. (2)

For any subgroup $H$ of $G$, the natural embedding of $\mathbb{C}[H]$ into
$\mathbb{C}[G]$ extends to an isometric unital $*$-homomorphism from
$\ell^{1}(H)$ into $\ell^{1}(G)$. We can thus identify $\ell^{1}(H)$ with a
unital Banach $*$-subalgebra of $\ell^{1}(G)$.

######  Proposition 6.19.

Let $H$ and $K$ be two distinct subgroups of a discrete group $G$. Then,

(6.5)
$d_{KK}\big{(}\mathbb{C}[H],\mathbb{C}[K]\big{)}=d_{KK}\Big{(}\ell^{1}(H),\ell^{1}(K)\Big{)}=1$

in $\ell^{1}(G)$.

###### Proof.

Since $\mathbb{C}[G]$ is dense in $\ell^{1}(G)$, by Lemma 3.3, it is enough to
show that $d_{KK}(\mathbb{C}[H]),\mathbb{C}[K])=1$ in $\ell^{1}(G)$. In view
of Remark 3.2(3), we can assume that $H\neq H\cap K\neq K$.

For convenience, let $C:=\mathbb{C}[H]$ and $D:=\mathbb{C}[K]$ and let $h\in
H\setminus H\cap K$. Then,

$\|h-\sum_{k\in K}\alpha_{k}k\|_{1}\geq 1$

for all $\sum_{k\in K}\alpha_{k}k\in B_{1}(D)$. This implies that,
$d(h,B_{1}(D))\geq 1$. Thus, by definition, we get

$d_{KK}(\mathbb{C}[H]),\mathbb{C}[K])\geq 1,$

and we are done. ∎

###  6.3. Kadison-Kastler and Christensen distances between subgroup von
Neumann subalgebras

Recall that for any discrete group $G$, the group von Neumann algebra
associated to $G$ is the von Neumann algebra given by

$L(G)=\\{\lambda(G)\\}^{\prime\prime}\subseteq B(\ell^{2}(G)),$

where $\lambda:G\to B(\ell^{2}(G))$ is the left regular (unitary)
representation of $G$. Also, there is a natural $*$-isomorhphism between
$\mathbb{C}[G]$ and $*$-$\text{alg}(\lambda(G))$; and, for any subgroup $H$ of
$G$, there is a natural $*$-isomorphism from $L(H)$ onto
$\lambda(H)^{\prime\prime}\subset L(G)$; thus, $L(H)$ can be considered as a
von Neumann subalgebra of $L(G)$. Further, $L(G)$ always admits a faithful
normal tracial state $\tau$ given by $\tau(x)=\langle
x(\delta_{e}),\delta_{e}\rangle$ for $x\in L(G)$. Thus, $L(G)$ admits an
inner-product structure via $\tau$ which induces a norm $\|\cdot\|_{\tau}$ on
$L(G)$ given by $\|x\|_{\tau}=\tau(x^{*}x)^{1/2}$, $x\in L(G)$.

######  Proposition 6.20.

Let $G$ be a discrete group and $H$ and $K$ be two distinct non-trivial
subgroups of $G$. Then, in $L(G)$, the distances

$d_{C}\big{(}\mathbb{C}[H],\mathbb{C}[K]\big{)},d_{MT}\big{(}\mathbb{C}[H],\mathbb{C}[K]\big{)},d_{KK}\big{(}\mathbb{C}[H],\mathbb{C}[K]\big{)},$
$d_{C}\big{(}L(H),L(K)\big{)},d_{MT}\big{(}L(H),L(K)\big{)}\text{ and
}d_{KK}\big{(}L(H),L(K)\big{)}$

are all equal to $1$.

###### Proof.

By Lemma 5.6, $d_{MT}\big{(}\mathbb{C}[H],\mathbb{C}[K]\big{)}\leq
d_{KK}\big{(}\mathbb{C}[H],\mathbb{C}[K]\big{)}\leq 1$. So, in view of
Proposition 5.5 and Proposition 5.8, it is enough to show that
$d_{MT}\big{(}\mathbb{C}[H],\mathbb{C}[K]\big{)}\geq 1$.

We prove this by considering the following two possibilities separately.

Case 1. Suppose that $H\subsetneq K$ or $K\subsetneq H$.

Without loss of generality, assume that $H\subsetneq K$. Then,
$\mathbb{C}[H]\subsetneq\mathbb{C}[K]$ and
$d_{\tau}\Big{(}\widehat{a},\widehat{B_{1}(\mathbb{C}[K])}\Big{)}=0$ for all
$a\in B_{1}(\mathbb{C}[H])$. Further, for any $k\in K\setminus H\cap K$, we
have, $\|\widehat{k}-\widehat{x}\|_{\tau}\geq 1$ for all $x\in
B_{1}(\mathbb{C}[H])$. Thus,

$\sup_{z\in
B_{1}(\mathbb{C}[K]))}d_{\tau}\Big{(}\widehat{z},\widehat{B_{1}(\mathbb{C}[H])}\Big{)}\geq
1;$

and, hence, $d_{MT}(\mathbb{C}[H],\mathbb{C}[K])\geq 1$.

Case 2. Suppose that $H\neq H\cap K\neq K$.

Then, again for any $h\in H\setminus H\cap K$, we have
$\|\widehat{h}-\widehat{x}\|_{\tau}\geq 1$ for all $x\in
B_{1}(\mathbb{C}[K])$. Thus, as above,
$d_{MT}(\mathbb{C}[H],\mathbb{C}[K])\geq 1$, and we are done. ∎

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