# Uniqueness of extremal almost periodic states on the injective type
$\mathrm{III}_{1}$ factor

Michael Hartglass∘ ∘Department of Mathematics and Computer Science, Santa
Clara University<EMAIL_ADDRESS>and  Brent Nelson∙ ∙Department of
Mathematics, Michigan State University<EMAIL_ADDRESS>
###### Abstract.

Let $R_{\infty}$ denote the Araki–Woods factor—the unique separable injective
type $\mathrm{III}_{1}$ factor. For extremal almost periodic states
$\varphi,\psi\in(R_{\infty})_{*}$, we show that if $\Delta_{\varphi}$ and
$\Delta_{\psi}$ have the same point spectrum then $\psi=\varphi\circ\alpha$
for some $\alpha\in\text{Aut}(R_{\infty})$. Consequently, the extremal almost
periodic states on $R_{\infty}$ are parameterized by countable dense subgroups
of $\mathbb{R}_{+}$, up to precomposition by automorphisms. As an application,
we show that KMS states for generalized gauge actions on Cuntz algebras agree
(up to an automorphism) with tensor products of Powers states on their von
Neumann completions.

## Introduction

Using approximations by completely positive maps, Haagerup provided a new
proof in [Haa89] of the uniqueness of the separable injective type
$\mathrm{III}_{\lambda}$ factors for $0<\lambda<1$. This uniqueness was first
established by Connes in his celebrated work [Con76], but a novel consequence
of Haagerup’s proof was the uniqueness of their
$\frac{2\pi}{\log(1/\lambda)}$-periodic states. For $0<\lambda\leq 1$ denote

$\displaystyle(R_{\lambda},\varphi_{\lambda}):=\bigotimes_{n\in\mathbb{N}}(M_{2}(\mathbb{C}),\phi_{\lambda})$
(1)

where

$\phi_{\lambda}(x):=\text{Tr}\left[\left(\begin{array}[]{cc}\frac{1}{1+\lambda}&0\\\
0&\frac{\lambda}{1+\lambda}\end{array}\right)x\right].$

These are separable injective factors of type $\mathrm{III}_{\lambda}$ for
$0<\lambda<1$ (also known as _Powers’ factors_ having been first studied in
[Pow67]), and type $\mathrm{II}_{1}$ for $\lambda=1$. The associated states
are $\frac{2\pi}{\log(1/\lambda)}$-periodic for $0<\lambda<1$, while
$\tau:=\varphi_{1}$ is of course the unique tracial state on $R:=R_{1}$. For
$(N,\varphi)$ a separable injective type $\mathrm{III}_{\lambda}$ factor
equipped with a $\frac{2\pi}{\log(1/\lambda)}$-periodic state and
$0<\lambda<1$, Haagerup established a state-preserving isomorphism
$(N,\varphi)\cong(R_{\lambda},\varphi_{\lambda})$, and consequently any pair
of $\frac{2\pi}{\log(1/\lambda)}$-periodic states on $R_{\lambda}$ differ only
by precomposition with an automorphism (see [Haa89, Theorem 6.1 and Corollary
6.5]). Haagerup also established the uniqueness of the separable injective
type $\mathrm{III}_{1}$ factor in [Haa87] using a reduction due to Connes in
[Con85], and a second proof was published posthumously in [Haa16] that more
closely followed the approach in [Haa89]. However, neither of these proofs
provide a _state preserving_ isomorphism and consequently the analogue of
[Haa89, Corollary 6.5] has been missing in the type $\mathrm{III}_{1}$ setting
until now. In this article we prove such an analogue, where periodic states
are replaced with so-called extremal almost periodic states.

In [Con74], Connes defined an _almost periodic_ weight as a faithful normal
semifinite weight $\varphi$ whose modular operator $\Delta_{\varphi}$ is
diagonalizable:

$\Delta_{\varphi}=\sum_{\lambda\in\text{Sd}(\varphi)}\lambda
1_{\\{\lambda\\}}(\Delta_{\varphi}),$

where we denote the point spectrum of $\Delta_{\varphi}$ by
$\text{Sd}(\varphi)$. An _extremal_ weight is one whose centralizer subalgebra
$M^{\varphi}:=\\{x\in M\colon\sigma_{t}^{\varphi}(x)=x\ \forall
t\in\mathbb{R}\\}$ is a factor. Almost periodic weights have the property that
$(M^{\varphi})^{\prime}\cap M=(M^{\varphi})^{\prime}\cap M^{\varphi}$ (see
[Con72, Theorem 10]), and so extremal almost periodic weights only occur on
factors. Additionally, $\text{Sd}(\varphi)$ for an extremal almost periodic
weight $\varphi$ is a subgroup of $\mathbb{R}_{+}$, the group of positive
reals $(0,\infty)$ under multiplication (see, for example, [Dyk97, Lemma
4.9]). The states $\varphi_{\lambda}$ in (1) are extremal almost periodic: for
$\lambda<1$ this follows from [Con73, Theorem 4.2.6]; and $\tau$ is tracial so
that $\Delta_{\tau}=1$ and $R^{\tau}=R$. Aperiodic examples of extremal almost
periodic weights necessarily occur on type $\mathrm{III}_{1}$ factors, and,
while not all type $\mathrm{III}_{1}$ factors admit such weights (see [Con74,
Corollary 5.3]), the unique separable injective type $\mathrm{III}_{1}$ factor
$R_{\infty}$ (also known as the _Araki–Woods factor_) has many. To see this,
extend the notation in (1) to $\lambda>1$ by letting
$\phi_{\lambda}:=\phi_{1/\lambda}$ and
$(R_{\lambda},\varphi_{\lambda}):=(R_{1/\lambda},\varphi_{1/\lambda})$. For a
countable subgroup $\Lambda\leq\mathbb{R}_{+}$, denote

$\displaystyle(R_{\Lambda},\varphi_{\Lambda}):=\bigotimes_{\lambda\in\Lambda}(R_{\lambda},\varphi_{\lambda}).$
(2)

If $\Lambda$ is of the form $\lambda^{\mathbb{Z}}$ for some $0<\lambda\leq 1$,
then $(R_{\Lambda},\varphi_{\Lambda})\cong(R_{\lambda},\varphi_{\lambda})$
thanks to the property
$(R_{\lambda}\bar{\otimes}R_{\lambda},\varphi_{\lambda}\otimes\varphi_{\lambda})=(R_{\lambda},\varphi_{\lambda})$.
Otherwise, $\Lambda$ is dense in $\mathbb{R}_{+}$ and $R_{\Lambda}\cong
R_{\infty}$. Moreover, $\varphi_{\Lambda}$ is an extremal almost periodic
state (see, for example, [HI22, Lemma 5.2.(i)]). Our main result is the
following:

######  Theorem A.

Let $N$ be a diffuse separable injective factor equipped with an extremal
almost periodic state $\varphi$. Then

$(N,\varphi)\cong(R_{\Lambda},\varphi_{\Lambda})$

where $\Lambda=\text{Sd}(\varphi)$.

If $\Lambda=\lambda^{\mathbb{Z}}$ for some $0<\lambda\leq 1$ then the above is
simply either [Haa89, Theorem 6.1] for $\lambda<1$ or [Con76, Theorem 1] for
$\lambda=1$. Thus the novelty of the above theorem lies only in the case that
$\Lambda$ is dense in $\mathbb{R}_{+}$, but nevertheless our proof
simultaneously covers all cases. The proof of Theorem A very closely follows
Haagerup’s proof of [Haa89, Theorem 6.1] and can be found in Section 5.
Roughly, the strategy is to approximate a countable generating set for $N$ by
a sequence of finite dimensional subfactors that when equipped with the
restriction of $\varphi$ are isomorphic to a finite dimensional tensor factor
of $(R_{\Lambda},\varphi_{\Lambda})$. These subfactors are constructed using
approximate completely positive factorizations of the identity map through
matrix algebras, whose existence is established in Section 3 (paralleling
[Haa89, Section 5]). Notably, Lemma 3.1 marks a technical deviation from
Haagerup’s ideas, though it still essentially corresponds to [Haa89, Lemma
5.3]. These completely positive maps also respect the modular theory of
$\varphi$, a property which is needed to apply the results in Section 4
(paralleling [Haa89, Section 4]). These results, namely Theorem 4.4,
approximate the completely positive maps by
$x\mapsto\sum_{n=1}^{\infty}a_{n}^{*}xa_{n}$ for
$(a_{n})_{n\in\mathbb{N}}\subset N^{\varphi}$ a partition of unity, which in
turn allows one to further approximate the maps by an inner automorphism using
the notion of $\delta$-related tuples from [Haa89, Section 2]. Despite how
similar our results are to those in [Haa89], working with non-cyclic subgroups
of $\mathbb{R}_{+}$ almost always prevents us from simply citing the
corresponding results. So instead the details are presented in full and the
reader should find the paper mostly self-contained.

We also remark that if one could directly argue that a separable injective
type $\mathrm{III}_{1}$ factor admits at least one extremal almost periodic
state, then Theorem A would yield a new proof of the uniqueness of such
factors. However, this remains out of reach. In fact, Haagerup’s approach in
[Haa87] specifically avoids this by working instead with dominant weights,
which are known to always exist for properly infinite factors by [CT77,
Chapter II].

As an immediate consequence of Theorem A we have the following.

######  Corollary B.

If $\varphi$ and $\psi$ are extremal almost periodic states on $R_{\infty}$
with $\text{Sd}(\varphi)=\text{Sd}(\psi)$, then there exists
$\alpha\in\text{Aut}(R_{\infty})$ satisfying $\psi=\varphi\circ\alpha$.

This corollary combined with the existence of the extremal almost periodic
states $\varphi_{\Lambda}$ in (2) implies there is a one-to-one correspondence
between the collection of countable dense subgroups of $\mathbb{R}_{+}$ and
the equivalence classes of extremal almost periodic states on $R_{\infty}$
under precomposition by automorphisms.

Due to its uniqueness, $R_{\infty}$ arises from a variety of constructions,
and so our results can be used to recognize extremal almost periodic states on
these constructions as some $\varphi_{\Lambda}$. Our last main result is an
example of this involving the Cuntz algebra. If $J$ is any finite or countably
infinite set, then recall that the Cuntz algebra, $\mathcal{O}_{J}$, is the
universal C*-algebra generated by a family of isometries $\\{v_{j}\\}_{j\in
J}$ with mutually orthogonal ranges, and if $J$ is finite it is further
required that $\sum_{j\in J}v_{j}v_{j}^{*}=1$. Let $\mu\colon J\to(0,1)$ be a
function with the property that $\sum_{j\in J}\mu(j)=1$. Define a gauge action
$\mathbb{R}\overset{\sigma}{\curvearrowright}\mathcal{O}_{J}$ by

$\sigma_{t}(v_{j})=\mu(j)^{it}v_{j},$

for each $t\in\mathbb{R}$ and $j\in J$. It is known that a unique KMS state
$\varphi$ exists for such an action (see, for example, [Tho17]). Letting
$\pi_{\varphi}$ be its GNS representation, we will abuse notation slightly and
let $\varphi$ also denote its extension to
$\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime}$ as a faithful normal state.
Denote by $\Lambda$ the multiplicative subgroup of $\mathbb{R}_{+}$ generated
by $\mu(J)$. It has been shown by Izumi for $J$ finite [Izu93] and Thomsen for
$J$ infinite [Tho19] that $\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime}$ is a
separable injective factor of type $\mathrm{III}_{\lambda}$ if
$\Lambda=\lambda^{\mathbb{Z}}$ for some $\lambda\in(0,1)$ and of type
$\mathrm{III}_{1}$ otherwise. In the former case, [Haa89, Theorem 6.1] implies
$(\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime},\varphi)\cong(R_{\lambda},\varphi_{\lambda})$
since $\varphi$ is necessarily $\frac{2\pi}{\log(1/\lambda)}$-periodic. In the
general case we argue in Section 6 that Theorem A can be applied to obtain the
following.

######  Theorem C.

With the above notation, $\varphi$ is an extremal almost periodic state on
$\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime}$ and there is a state
preserving isomorphism
$(\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime},\varphi)\cong(R_{\Lambda},\varphi_{\Lambda})$.

### Acknowledgements

The authors thank Cyril Houdayer for suggesting this problem. The second
author was supported by NSF grant DMS-2247047.

## 1\. Preliminaries

We say a von Neumann algebra $M$ is _separable_ if it is separable in the
weak* topology induced by $M\cong(M_{*})^{*}$; equivalently, if $M_{*}$ is
separable as a Banach space. We write $N\leq M$ when $N$ is a unital von
Neumann subalgebra of $M$. For a family of subalgebras $\\{N_{i}\leq M\colon
i\in I\\}$, we write $\bigvee_{i\in I}N_{i}$ for the von Neumann algebra
generated by $\bigcup_{i\in I}N_{i}$. For a state $\varphi$ on $M$ we define
the inner product

$\left\langle x,y\right\rangle_{\varphi}=\varphi(y^{*}x)\qquad x,y\in M,$

and denote its associated norm by $\|x\|_{\varphi}$.

We assume the reader has some familiarity with weights and modular theory, but
the complete details can be found in [Tak03] (see also [GLN24, Section 1]).
For a faithful normal semifinite weight on $M$, its _modular automorphism
group_ refers to the action
$\mathbb{R}\overset{\sigma^{\varphi}}{\curvearrowright}M$ defined by

$\sigma_{t}^{\varphi}(x):=\Delta_{\varphi}^{it}x\Delta_{\varphi}^{-it}.$

We say a subalgebra $N\leq M$ is _$\varphi$ -invariant_ if $\varphi|_{N}$ is
semifinite and $\sigma_{t}^{\varphi}(N)=N$ for all $t\in\mathbb{R}$, which is
equivalent to the existence of a faithful normal $\varphi$-invariant
conditional expectation from $M$ onto $N$ (see [Tak03, Theorem IX.4.2]). For
$\lambda\in\mathbb{R}_{+}$ we denote

$M^{(\varphi,\lambda)}:=\\{x\in M\colon\sigma_{t}^{\varphi}(x)=\lambda^{it}x\
\forall t\in\mathbb{R}\\},$

so that $M^{(\varphi,1)}=M^{\varphi}$. We call such elements _eigenoperators_
of $\varphi$. We also denote

$M^{(\varphi,\text{eig})}:=\text{span}\left(\bigcup_{\lambda\in\mathbb{R}_{+}}M^{(\varphi,\lambda)}\right),$

which is a unital $*$-subalgebra of $M$, and it is weak* dense when $\varphi$
is almost periodic (see [GLN24, Lemma 1.4]).

Given an almost periodic weight $\varphi$ on $M$, let
$\Lambda\leq\mathbb{R}_{+}$ be the group generated by $\text{Sd}(\varphi)$.
Equipping $\Lambda$ with the discrete topology, let $G$ denote its compact
dual group. Identifying the dual of $\mathbb{R}_{+}$ with $\mathbb{R}$ via the
pairing $(s,t)=s^{it}$, the _transpose_ of the inclusion map
$\iota\colon\Lambda\to\mathbb{R}_{+}$ gives a continuous group homomorphsim
$\hat{\iota}\colon\mathbb{R}\to G$ determined by

$(\hat{\iota}(t)|\lambda)=(\iota(\lambda)|t)=\lambda^{it}.$

The action $\mathbb{R}\overset{\sigma^{\varphi}}{\curvearrowright}M$ given by
the modular automorphism group of $\varphi$ determines an action
$G\overset{\alpha}{\curvearrowright}M$ satisfying
$\alpha_{\hat{\iota}(t)}=\sigma_{t}^{\varphi}$ for all $t\in\mathbb{R}$. In
particular, $M^{\alpha}=M^{\varphi}$ and more generally for $x\in
M^{(\varphi,\lambda)}$ one has $\alpha_{s}(x)=(\lambda|s)x$ for all $s\in G$.
We call this action of $G$ the _point modular extension_ of
$\mathbb{R}\overset{\sigma^{\varphi}}{\curvearrowright}M$. For each
$\lambda\in\Lambda$, there is a normal $M^{\varphi}$-$M^{\varphi}$-bimodular
idempotent map $\mathcal{E}_{(\varphi,\lambda)}\colon M\to
M^{(\varphi,\lambda)}$ defined by

$\mathcal{E}_{(\varphi,\lambda)}(x):=\int_{G}\overline{(\lambda|s)}\alpha_{s}(x)\
d\mu(s),$

where $\mu$ is a normalized Haar measure on $G$. In particular,
$\mathcal{E}_{\varphi}:=\mathcal{E}_{(\varphi,1)}$ is the unique
$\varphi$-preserving faithful normal conditional expectation onto the
centralizer. If $\varphi$ is a state, then one also has

$\|x\|_{\varphi}^{2}=\sum_{\lambda\in\Lambda}\|\mathcal{E}_{(\varphi,\lambda)}(x)\|_{\varphi}^{2},$

for all $x\in M$ since
$\Delta_{\varphi}=\sum_{\lambda\in\Lambda}1_{\\{\lambda\\}}(\Delta_{\varphi})$.
See [Dyk95, Dyk97] for additional details.

Suppose that $F$ is a finite-dimensional subfactor of a factor $N$, and
$F^{c}$ is defined by $F^{c}=F^{\prime}\cap N$. Note that this means $N\cong
F\otimes F^{c}$. For a faithful normal state $\varphi$ on $N$, the following
three conditions are equivalent (see, for example, [Haa89, Section 4]):

  1. (1)

$F$ is $\Delta_{\varphi}$-invariant;

  2. (2)

there exists a $\varphi$-preserving conditional expectation from $N$ onto $F$;

  3. (3)

$(N,\varphi)\cong(F,\varphi)\otimes(F^{c},\varphi)$.

For the purposes of this article, condition (3) above will be the one which is
easiest to exploit. We will denote this more compactly as
$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}$.

## 2\. Extremal almost periodic weights

In this section we present a few ways to construct new extremal almost
periodic weights out of old ones. We will only need to apply them to _states_
in the present article, but we prove them in the generality of weights
whenever possible because they may be of independent interest. The following
lemma is proven in [HI22, Lemma 5.2.(ii)] in the case of states and for
$M=R_{\infty}$ the Araki–Woods factor. However, the proof in the generality
below is exactly the same.

######  Lemma 2.1.

Let $(M,\varphi)$ be a von Neumann algebra equipped with an extremal almost
periodic weight. If $(N,\psi)$ is a factor equipped with an almost periodic
weight satisfying $\text{Sd}(\psi)\subset\text{Sd}(\varphi)$, then
$\varphi\otimes\psi$ is extremal almost periodic on $M\bar{\otimes}N$ with
$\text{Sd}(\varphi\otimes\psi)=\text{Sd}(\varphi)$.

The following can be viewed as a partial converse to the previous lemma.

######  Lemma 2.2.

Let $M=P\bar{\otimes}N$ be factor with $P$ a type $\mathrm{I}$ factor. Suppose
$\varphi$ is an extremal almost periodic weight on $M$ satisfying:
$\varphi|_{P}$ and $\varphi|_{N}$ are seminifinite; $\varphi|_{P}$ is almost
periodic; and $\varphi=\varphi|_{P}\otimes\varphi|_{N}$. Then $\varphi|_{N}$
is extremal almost periodic with $\text{Sd}(\varphi|_{N})=\text{Sd}(\varphi)$.
In particular, $\text{Sd}(\varphi|_{P})\subset\text{Sd}(\varphi|_{N})$.

###### Proof.

Identifying $P\cong B(\mathcal{H})$ for some Hilbert space $\mathcal{H}$, we
have that $\varphi|_{P}(x)=\text{Tr}(h^{\frac{1}{2}}xh^{\frac{1}{2}})$ for
some positive (unbounded) operator $h$ on $\mathcal{H}$ and
$\sigma_{t}^{\varphi|_{P}}(x)=h^{it}xh^{-it}$ for all $t\in\mathbb{R}$. Since
$\varphi|_{P}$ is almost periodic, $h$ is necessarily diagonalizable with
respect to some family of matrix units $\\{e_{i,j}\in B(\mathcal{H})\colon
i,j\in I\\}$, In particular one has $e_{i,i}\otimes 1\in M^{\varphi}$ for all
$i\in I$. Because $(N,\varphi|_{N})$ can be identified with $((e_{i,i}\otimes
1)M(e_{i,i}\otimes 1),\frac{1}{\text{Tr}(he_{i,i})}\varphi)$ for any $i\in I$,
it follows from [GLN24, Remark 1.5] that $\varphi|_{N}$ is extremal almost
periodic.

Next, we see that $e_{i,j}$ is an eigenoperator of $\varphi|_{P}$ for each
$i,j\in I$ since $h$ is diagonalized with respect to these matrix units, say
with eigenvalue $\lambda_{i,j}$. Note that $\lambda_{j,i}=1/\lambda_{i,j}$.
Using that $M^{\varphi}$ is a factor we have
$e_{i,i}M^{\varphi}e_{j,j}=e_{i,j}\otimes N$ is non-zero for each $i,j\in I$.
Hence there is a non-zero $x\in N$ with $e_{i,j}\otimes x\in M^{\varphi}$, and
the factorization $\varphi=\varphi|_{P}\otimes\varphi|_{N}$ implies $x\in
N^{(\varphi|_{N},1/\lambda_{i,j})}=N^{(\varphi|_{N},\lambda_{j,i})}$.
Consequently,

$\text{Sd}(\varphi|_{P})=\\{\lambda_{i,j}\colon i,j\in
I\\}\subset\text{Sd}(\varphi|_{N}).$

Since $\text{Sd}(\varphi|_{N})$ is a group and
$\text{Sd}(\varphi)=\text{Sd}(\varphi|_{P})\text{Sd}(\varphi|_{N})$, it
follows that $\text{Sd}(\varphi|_{N})=\text{Sd}(\varphi)$ as claimed. ∎

Observe that if $\varphi$ in previous lemma is a state, then of course its
restrictions to $P$ and $N$ are automatically semifinite, but also
$\varphi|_{P}=\text{Tr}(h^{\frac{1}{2}}\,\cdot\,h^{\frac{1}{2}})$ is
automatically almost periodic since $h$ is necessarily trace class and hence
diagonalizable.

######  Lemma 2.3.

Let $M$ be a factor with separable predual equipped with an extremal almost
periodic weight $\varphi$, let $\\{p_{i}\in M^{\varphi}\colon i\in I\\}$ be a
family of projections satisfying $\sum_{i}p_{i}=1$, and let
$\\{\mu_{i}\in\mathbb{R}_{+}\colon i\in I\\}$ satisfy
$\frac{\mu_{i}}{\mu_{j}}\in\text{Sd}(\varphi)$ for all $i,j\in I$. Set

$h:=\sum_{i\in I}\mu_{i}p_{i}.$

Then $\psi:=\varphi(h^{\frac{1}{2}}\,\cdot\,h^{\frac{1}{2}})$ is an extremal
almost periodic weight with $\text{Sd}(\psi)=\text{Sd}(\varphi)$.

###### Proof.

The weight $\psi$ is faithful since $h$ is non-singular, and since $\varphi$
is strictly semifinite and $p_{i}M^{\varphi}p_{i}\subset M^{\psi}$ it follows
that $\psi$ is also strictly semifinite. Observe that $M^{(\psi,\text{eig})}$
contains $\text{span}\\{p_{i}M^{(\varphi,\text{eig})}p_{j}\colon i,j\in I\\}$
and is therefore weak* dense. Consequently, $\psi$ is almost periodic by
[GLN24, Lemma 1.4].

We next compute $\text{Sd}(\psi)$. Denote $\Lambda:=\text{Sd}(\varphi)$, which
is a group by virtue of $\varphi$ being extremal. Fix $\lambda\in\Lambda$ and
$i\in I$ and note that there exists a partial isometry $v\in
M^{(\varphi,\lambda)}$. Since $v^{*}v$ and $p_{i}$ are not centrally
orthogonal in $M^{\varphi}$, we can find a non-zero partial isometry $w\in
M^{\varphi}$ satisfying $w^{*}w\leq p_{i}$ and $ww^{*}\leq v^{*}v$. Hence
$v^{*}vwp_{i}=w$, and in particular $vwp_{i}$ is a non-zero partial isometry.
Since $vwp_{i}w^{*}v^{*}$ and $p_{i}$ are also not centrally orthogonal in
$M^{\varphi}$, we can find another partial isometry $u\in M^{\varphi}$
satisfying $u^{*}u\leq vwp_{i}w^{*}v^{*}$ and $uu^{*}\leq p_{i}$. Hence
$p_{i}uvwp_{i}w^{*}v^{*}=u$, and in particular $p_{i}uvwp_{i}\neq 0$. Also,
the Connes cocycle derivative theorem implies

$\sigma_{t}^{\psi}(p_{i}uvwp_{i})=h^{it}\sigma_{t}^{\varphi}(p_{i}uvwp_{i})h^{-it}=\mu_{i}^{it}p_{i}u\sigma_{t}^{\varphi}(v)wp_{i}\mu_{i}^{-it}=\lambda^{it}p_{i}uvwp_{i}.$

Hence $\Lambda\subset\text{Sd}(\psi)$. The reverse inclusion follows from
$\text{span}\\{p_{i}M^{(\varphi,\text{eig})}p_{j}\colon i,j\in I\\}$ being
weakly dense: $\frac{\mu_{i}}{\mu_{j}}\in\Lambda$ implies
$p_{i}M^{(\varphi,\lambda)}p_{j}\in\bigcup_{\nu\in\Lambda}M^{(\psi,\nu)}$.

Finally, we establish extremality. By replacing $\psi$ with
$\frac{1}{\mu_{i_{0}}}\psi$ for some fixed $i_{0}\in I$, we may assume
$\mu_{i}\in\Lambda$ for all $i\in I$. Let $G$ denote the compact dual to
$\Lambda$ (which we give the discrete topology) and let
$G\overset{\alpha}{\curvearrowright}M$ and
$G\overset{\beta}{\curvearrowright}M$ be the point modular extensions of
$\mathbb{R}\overset{\sigma^{\varphi}}{\curvearrowright}M$ and
$\mathbb{R}\overset{\sigma^{\psi}}{\curvearrowright}M$, respectively. Consider
the strongly continuous unitary representation

$G\ni s\mapsto u_{s}:=\sum_{i\in I}(s|\mu_{i})p_{i}\in M^{\varphi}.$

Then for $x\in M^{(\varphi,\lambda)}$ and $i,j\in I$ we have
$\beta_{s}(p_{i}xp_{j})=(s|\frac{\mu_{i}}{\mu_{j}}\lambda)p_{i}xp_{j}$ and

$u_{s}\alpha_{s}(p_{i}xp_{j})u_{s}^{*}=(s|\mu_{i})p_{i}\alpha_{s}(x)p_{j}(s|\mu_{j})^{-1}=(s|\frac{\mu_{i}}{\mu_{j}}\lambda)p_{i}xp_{j}.$

It follows that $\beta_{s}(x)=u_{s}\alpha_{s}(x)u_{s}^{*}$ for all $x\in M$
and hence $\alpha$ and $\beta$ are cocycle equivalent. Consequently,
$M\rtimes_{\alpha}G\cong M\rtimes_{\beta}G$ by [Tak03, Theorem X.1.7.(ii)]. By
[GLN24, Lemma 2.27], the former is a factor since $M^{\alpha}=M^{\varphi}$ is
a factor. Thus $M\rtimes_{\beta}G$ is a factor and we saw above that
$\text{Sd}(\psi)=\Lambda$, so the same lemma implies $M^{\beta}=M^{\psi}$ is a
factor. ∎

The next lemma is a generalization of [Haa89, Lemma 5.4]) and will be needed
in Section 3.

######  Lemma 2.4.

Let $M$ be a diffuse factor equipped with an extremal almost periodic state
$\varphi$, and let $(R,\tau)$ be the unique separable injective type
$\mathrm{II}_{1}$ factor equipped with its unique tracial state. For $h$
diagonlizable and affiliated with $R$, suppose
$\psi=\tau(h^{\frac{1}{2}}\,\cdot\,h^{\frac{1}{2}})$ is an almost periodic
state on $R$ with $\text{Sd}(\psi)\subset\text{Sd}(\varphi)$. Then there
exists a $\varphi$-invariant subfactor $P\leq M$ such that

$(R,\psi)\cong(P,\varphi|_{P}).$

###### Proof.

Let

$h=\sum_{i\in I}\mu_{i}p_{i}$

be the diagonalization of $h$, so that $p_{i}\in R$ for each $i\in I$ and
$\text{Sd}(\psi)=\\{\frac{\mu_{i}}{\mu_{j}}\colon i,j\in I\\}$. Since
$M^{\varphi}$ is a type $\mathrm{II}_{1}$ factor (see [GLN24, Lemma 3.1]), we
can find a partition of unity $\\{q_{i}\colon i\in I\\}\subset M^{\varphi}$
satisfying $\varphi(q_{i})=\psi(p_{i})=\mu_{i}\tau(p_{i})$ for all $i\in I$.
Set

$k:=\sum_{i\in I}\mu_{i}q_{i}$

and define $\omega:=\varphi(k^{-\frac{1}{2}}\,\cdot\,k^{-\frac{1}{2}})$. Then
$\omega$ is an extremal almost periodic weight on $M$ by Lemma 2.3. In fact,
$\omega$ is a state because

$\omega(1)=\sum_{i\in I}\frac{1}{\mu_{i}}\varphi(q_{i})=\sum_{i\in
I}\tau(p_{i})=1.$

Thus $M^{\omega}$ is a type $\mathrm{II}_{1}$ factor, and so we can find an
embedding $\alpha\colon R\to M^{\omega}$ and set $P:=\alpha(M)$. Conjugating
by a unitary in $M^{\omega}$ if necessary, we may assume $q_{i}\in P$ for each
$i\in I$. We have $\omega\circ\alpha=\tau$ by uniqueness of the trace on $R$,
so in particular for $f_{i}:=\alpha(p_{i})$ we have

$\omega(f_{i})=\tau(p_{i})=\omega(q_{i}).$

Thus $f_{i}$ is equivalent to $q_{i}$ in $P$, so let $u_{i}\in P$ be a partial
isometry satisfying $u_{i}^{*}u_{i}=f_{i}$ and $u_{i}u_{i}^{*}=q_{i}$ and set
$u:=\sum_{i\in I}u_{i}$. Then $u$ is a unitary satisfying $uf_{i}u^{*}=q_{i}$
for all $i\in I$, and hence $\beta:=\text{Ad}\left(u\right)\circ\alpha$ is a
unital $*$-isomorphism from $R$ to $P$ that further satisfies

$\beta(h)=\sum_{i\in I}\mu_{i}u\alpha(p_{i})u^{*}=\sum_{i\in
I}\mu_{i}q_{i}=k.$

Consequently, for $x\in R$ we have

$\varphi(\beta(x))=\omega(k^{\frac{1}{2}}\beta(x)k^{\frac{1}{2}})=\omega(\beta(h^{\frac{1}{2}}xh^{\frac{1}{2}}))=\omega(\alpha(h^{\frac{1}{2}}xh^{\frac{1}{2}}))=\tau(h^{\frac{1}{2}}xh^{\frac{1}{2}})=\psi(x),$

where have used $u\in P\subset M^{\omega}$ in the third equality. Therefore
$(R,\psi)\cong(P,\varphi|_{P})$. Finally, note that the Connes cocycle
derivative theorem implies
$\sigma_{t}^{\varphi}(x)=k^{it}\sigma_{t}^{\omega}(x)k^{-it}$, and since
$k^{it}\in P$ for all $t\in\mathbb{R}$ it follows that $P$ is
$\varphi$-invariant. ∎

Let $(M,\varphi)$ and $(R,\tau)$ be as in the pervious lemma. Suppose $\psi$
is a faithful state on $M_{d}(\mathbb{C})$ with
$\text{Sd}(\psi)\subset\text{Sd}(\varphi)$. Then
$\psi=\text{tr}_{d}(h^{\frac{1}{2}}\,\cdot\,h^{\frac{1}{2}})$ for some
necessarily diagonalizable positive definite matrix $h\in M_{d}(\mathbb{C})$,
where $\text{tr}_{d}$ is the normalized trace. Since
$(M_{d}(\mathbb{C})\otimes R,\text{tr}_{d}\otimes\tau)\cong(R,\tau)$, we can
apply the previous lemma to $\psi\otimes\tau$ to obtain the following almost
periodic analogue of [Haa89, Proposition 4.6]. Note that
$M_{d}(\mathbb{C})\cong M_{d}(\mathbb{C})\otimes\mathbb{C}$ is a
$(\psi\otimes\tau)$-invariant subfactor of $M_{d}(\mathbb{C})\otimes R$, and
thus its image in $P$ will be $\varphi$-invariant.

######  Lemma 2.5.

Let $M$ be a diffuse factor equipped with an extremal almost periodic state
$\varphi$. For $d\in\mathbb{N}$ let $\psi$ be a faithful state on
$M_{d}(\mathbb{C})$ with $\text{Sd}(\psi)\subset\text{Sd}(\varphi)$. Then
there exists a $\varphi$-invariant subfactor $F\leq M$ such that

$(M_{d}(\mathbb{C}),\psi)\cong(F,\varphi|_{F}).$

## 3\. Modular completely positive factorizations through matrix algebras

Given a $(N,\varphi)$, a diffuse separable injective factor equipped with an
extremal almost periodic state, we show in this section that the identity map
on $N$ can be pointwise approximated in $\|\cdot\|_{\varphi}$-norm by a
composition of completely maps to and from matrix algebras, which also respect
the modular theory of $\varphi$. This result, Thoerem 3.2, corresponds to
[Haa89, Theorem 5.1] and will be needed in the proof of Theorem A.

As an initial step, we first show such an approximations exist for completely
positive maps to and from the unique separable injective type
$\mathrm{II}_{1}$ factor $R$. The following lemma, which corresponds to
[Haa89, Lemma 5.3], is where the hypothesis of almost periodicity is most
heavily used in the form of the point modular extension of the modular
automorphism group of $\varphi$ and the existence of the idempotent maps
$\mathcal{E}_{(\varphi,\lambda)}$ for $\lambda\in\text{Sd}(\varphi)$.

######  Lemma 3.1.

Let $N$ be a diffuse separable injective factor, let $\varphi$ be an extremal
almost periodic state on $N$, and let $(R,\tau)$ be the unique separable
injective type $\rm{II}_{1}$ factor equipped with its unique tracial state.
For every finite set $x_{1},\ldots,x_{n}\in N$, finite set
$E\subset\text{Sd}(\varphi)$, and $\epsilon>0$, there exist normal unital
completely positive maps

$S\colon N\to R\qquad\text{ and }\qquad T\colon R\to N,$

and a faithful state $\psi=\tau(a^{\frac{1}{2}}\,\cdot\,a^{\frac{1}{2}})$ on
$R$ with $E\subset\text{Sd}(\psi)\subset\text{Sd}(\varphi)$, where $a\in
R_{+}$ has finite spectrum. Moreover, one has

$\displaystyle\psi\circ S$ $\displaystyle=\varphi,$ $\displaystyle\varphi\circ
T$ $\displaystyle=\psi,$ $\displaystyle\sigma_{t}^{\psi}\circ S$
$\displaystyle=S\circ\sigma_{t}^{\varphi},$
$\displaystyle\sigma_{t}^{\varphi}\circ T$
$\displaystyle=T\circ\sigma_{t}^{\psi},$

for all $t\in\mathbb{R}$, and

$\|T\circ S(x_{k})-x_{k}\|_{\varphi}<\epsilon\qquad k=1,\ldots,n.$

###### Proof.

Enlarging $E$ if necessary, we may assume that

$\displaystyle\left\|\sum_{\gamma\in
E}\mathcal{E}_{(\varphi,\gamma)}(x_{k})-x_{k}\right\|_{\varphi}<\frac{\epsilon}{2},$
(3)

for each $k=1,\ldots,n$. The multiplicative group generated by $E$ is finitely
generated and torsion free, so it is free abelian and therefore has a basis.
Let $\lambda_{1},\ldots,\lambda_{d}\subset Sd(\varphi)$ be such such a basis.
That is, $E\subset\\{\lambda_{1}^{r_{1}}\cdots\lambda_{d}^{r_{d}}\colon
r_{1},\ldots,r_{d}\in\mathbb{Z}\\}$ and
$\lambda_{1}^{r_{1}}\cdots\lambda_{d}^{r_{d}}=\lambda_{1}^{s_{1}}\cdots\lambda_{d}^{s_{d}}$
if and only if $r_{1}=s_{1},\ldots,r_{d}=s_{d}$.

Now, for $p\in\mathbb{N}$ consider the density matrix

$h:=\frac{1}{\sum_{r_{1},\ldots,r_{d}=1}^{p}\lambda_{1}^{r_{1}}\cdots\lambda_{d}^{r_{d}}}\sum_{r_{1},\ldots,r_{d}=1}^{p}\lambda_{1}^{r_{1}}\cdots\lambda_{d}^{r_{d}}e_{r_{1}r_{1}}\otimes\cdots\otimes
e_{r_{d}r_{d}}\in M_{p}(\mathbb{C})^{\otimes d},$

and let $\omega:=\text{Tr}(h\,\cdot\,)$ be the associated state on
$M_{p}(\mathbb{C})^{\otimes d}$. Observe that

$\sigma_{t}^{\omega}(e_{r_{1}s_{1}}\otimes\cdots\otimes
e_{r_{d}s_{d}})=\lambda_{1}^{i(r_{1}-s_{1})t}\cdots\lambda_{d}^{i(r_{d}-s_{d})t}e_{r_{1}s_{1}}\otimes\cdots\otimes
e_{r_{d}s_{d}}.$

Hence

$\text{Sd}(\omega)=\\{\lambda_{1}^{r_{1}}\cdots\lambda_{d}^{r_{d}}\colon-p<r_{1},\ldots,r_{d}<p\\}\subset
Sd(\varphi).$

Lemma 2.1 then implies that the centralizer of $N\otimes
M_{p}(\mathbb{C})^{\otimes d}$ with respect to $\chi:=\varphi\otimes\omega$
isomorphic to $R$ and $\text{Sd}(\chi)=\text{Sd}(\varphi)$. Equipping
$\text{Sd}(\varphi)$ with the discrete topology, let $G$ be its compact dual
group with normalized Haar measure $\mu$, and let
$G\overset{\alpha}{\curvearrowright}N$ and
$G\overset{\beta}{\curvearrowright}N\otimes M_{p}(\mathbb{C})^{\otimes d}$ be
the point modular extensions
$\mathbb{R}\overset{\sigma^{\varphi}}{\curvearrowright}N$ and
$\mathbb{R}\overset{\sigma^{\chi}}{\curvearrowright}N\otimes
M_{p}(\mathbb{C})^{\otimes d}$, respectively. Then

$\beta_{t}(x\otimes e_{r_{1}s_{1}}\otimes\cdots\otimes
e_{r_{d}s_{d}})=(t\mid\lambda_{1}^{r_{1}-s_{1}}\cdots\lambda_{d}^{r_{d}-s_{d}})\alpha_{t}(x)\otimes
e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}},$

so that

$\displaystyle(N$ $\displaystyle\otimes M_{p}(\mathbb{C})^{\otimes d})^{\chi}$
$\displaystyle=$
$\displaystyle\left\\{\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}x_{r_{1},s_{1},\ldots
r_{d},s_{d}}\otimes e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}}\in
N\otimes M_{p}(\mathbb{C})^{\otimes d}\colon x_{r_{1},s_{1},\ldots
r_{d},s_{d}}\in
N^{(\varphi,\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{d}-r_{d}})}\right\\}$

Also observe that for $\\{x_{r_{1},s_{1},\ldots r_{d},s_{d}}\in N\colon 1\leq
r_{1},s_{1}\ldots,r_{d},s_{d}\leq p\\}$ one has

$\displaystyle\mathcal{E}_{\chi}$
$\displaystyle\left(\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}x_{r_{1},s_{1},\ldots
r_{d},s_{d}}\otimes e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}}\right)$
$\displaystyle=\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}\int_{G}(\lambda_{1}^{r_{1}-s_{1}}\cdots\lambda_{d}^{r_{d}-s_{d}}\mid
t)\alpha_{t}(x_{r_{1},s_{1},\ldots r_{d},s_{d}})\otimes
e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}}\ d\mu(t)$
$\displaystyle=\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}\int_{G}\overline{(\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{r}-r_{d}}\mid
t)}\alpha_{t}(x_{r_{1},s_{1},\ldots r_{d},s_{d}})\otimes
e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}}\ d\mu(t)$
$\displaystyle=\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}\mathcal{E}_{(\varphi,\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{d}-r_{d}})}(x_{r_{1},s_{1},\ldots
r_{d},s_{d}})\otimes e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}}.$

Now, identifying $R\cong(N\otimes M_{p}(\mathbb{C})^{\otimes d})^{\chi}$ we
define $S\colon N\to R$ and $T\colon R\to N$ by

$S(x):=\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}\mathcal{E}_{(\varphi,\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{d}-r_{d}})}(x)\otimes
e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}}$

for $x\in N$, and

$T\left(\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}x_{r_{1},s_{1},\ldots
r_{d},s_{d}}\otimes e_{r_{1}s_{1}}\otimes\cdots\otimes
e_{r_{d}s_{d}}\right):=\frac{1}{p^{d}}\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}x_{r_{1},s_{1},\ldots
r_{d},s_{d}}$

for $x_{r_{1},s_{1},\ldots r_{d},s_{d}}\in
N^{(\varphi,\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{d}-r_{d}})}$. Then
$S$ and $T$ are unital by definition, and if we denote

$e_{0}:=\frac{1}{p^{d}}\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}e_{r_{1}s_{1}}\otimes\cdots\otimes
e_{r_{d}s_{d}},$

then $S=p^{d}\mathcal{E}_{\chi}(\,\cdot\,\otimes e_{0})$ and
$T=(\text{id}_{N}\otimes\text{Tr})((1\otimes e_{0})\,\cdot\,)$ are normal and
completely positive. Moreover, for each $x\in N$ we have

$T\circ
S(x)=\frac{1}{p^{d}}\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}\mathcal{E}_{(\varphi,\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{d}-r_{d}})}(x)=\sum_{|k_{1}|,\ldots,|k_{d}|<p}\left(1-\frac{|k_{1}|}{p}\right)\cdots\left(1-\frac{|k_{d}|}{p}\right)\mathcal{E}_{(\varphi,\lambda_{1}^{k_{1}}\cdots\lambda_{d}^{k_{d}})}(x).$

Suppose $p$ is large enough so that

$E\subset\\{\lambda_{1}^{k_{1}}\cdots\lambda_{d}^{k_{d}}\colon|k_{1}|,\ldots,|k_{d}|<p\\}.$

If we denote $y_{k}:=\sum_{\gamma\in E}\mathcal{E}_{(\varphi,\gamma)}(x_{k})$
for each $k=1,\ldots,n$, then we have

$\displaystyle\|T\circ S(x_{k})-x_{k}\|_{\varphi}$ $\displaystyle\leq\|T\circ
S(x_{k}-y_{k})\|_{\varphi}+\|T\circ
S(y_{k})-y_{k}\|_{\varphi}+\|y_{k}-x_{k}\|_{\varphi}$
$\displaystyle<\epsilon+\left\|\sum_{\lambda_{1}^{k_{1}}\cdots\lambda_{d}^{k_{d}}\in
E}\left[1-\left(1-\frac{|k_{1}|}{p}\right)\cdots\left(1-\frac{|k_{d}|}{p}\right)\right]\mathcal{E}_{(\varphi,\lambda_{1}^{k_{1}}\cdots\lambda_{d}^{k_{d}})}(x_{k})\right\|_{\varphi},$

where we have used (3) and the fact that $\|T\circ
S(x)\|_{\varphi}\leq\|x\|_{\varphi}$ (which follows from $T\circ S\colon N\to
N$ being a unital completely positive map). Thus for sufficiently large $p$
one has

$\|T\circ S(x_{k})-x_{k}\|_{\varphi}<\epsilon$

for each $k=1,\ldots,n$.

Finally, set $\psi:=\varphi\circ T$. Then for $x\in N$ one has

$\psi\circ S(x)=\varphi\circ T\circ
S(x)=\frac{1}{p^{d}}\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}\varphi\circ\mathcal{E}_{(\varphi,\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{d}-r_{d}})}(x)=\frac{1}{p^{d}}\sum_{r_{1},\ldots,r_{d}=1}^{p}\varphi(x)=\varphi(x).$

Similarly, for $x_{r_{1},s_{1},\ldots r_{d},s_{d}}\in
N^{(\varphi,\lambda_{1}^{s_{1}-r_{1}}\cdots\lambda_{d}^{s_{d}-r_{d}})}$ one
has

$\displaystyle\psi$
$\displaystyle\left(\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}x_{r_{1},s_{1},\ldots
r_{d},s_{d}}\otimes e_{r_{1}s_{1}}\otimes\cdots\otimes e_{r_{d}s_{d}}\right)$
$\displaystyle\qquad=\frac{1}{p^{d}}\sum_{r_{1},s_{1},\ldots,r_{d},s_{d}=1}^{p}\varphi(x_{r_{1},s_{1},\ldots
r_{d},s_{d}})=\frac{1}{p^{d}}\sum_{r_{1},\ldots,r_{d}=1}^{p}\varphi(x_{r_{1},r_{1},\ldots
r_{d},r_{d}}).$

Hence $\psi=\frac{1}{p^{d}}(\varphi\otimes\text{Tr})$. Note that this implies
$\sigma_{t}^{\psi}=\sigma^{\varphi}_{t}\otimes\text{id}$, and thus by
definition of $S$ and $T$ one has

$\displaystyle\sigma_{t}^{\psi}\circ S=S\circ\sigma_{t}^{\varphi}$
$\displaystyle\sigma_{t}^{\varphi}\circ T=T\circ\sigma_{t}^{\psi}.$

Recalling that $\chi=\varphi\otimes\omega=(\varphi\otimes\text{Tr})((1\otimes
h)\,\cdot\,)$, we see that if

$a:=\frac{1}{p^{d}}(1\otimes h^{-1}),$

then $\psi=\chi(a\,\cdot\,)$. Since $(R,\tau)\cong((N\otimes
M_{p}(\mathbb{C})^{\otimes d})^{\chi},\chi)$, this implies $\text{Sd}(\psi)$
is the finite set consisting of the ratios of eigenvalues of $h$ and so
$E\subset\text{Sd}(\psi)\subset\text{Sd}(\varphi)$. ∎

We require some additional notation for the proof of the main theorem in this
section. For a finite subset $E\subset\mathbb{R}_{+}$ we denote

$\displaystyle(\mathbb{M}_{E},\phi_{E})$
$\displaystyle:=\bigotimes_{\lambda\in E}(M_{2}(\mathbb{C}),\phi_{\lambda})$
$\displaystyle(R_{E},\varphi_{E})$
$\displaystyle:=\overline{\bigotimes_{\lambda\in
E}}(R_{\lambda},\varphi_{\lambda}).$

Note that $\varphi_{E}$ is extremal almost periodic and

$\displaystyle\text{Sd}(\phi_{E})=\prod_{\lambda\in
E}\text{Sd}(\phi_{\lambda})=\left\\{\prod_{\lambda\in
E}\lambda^{e(\lambda)}\colon e(\lambda)\in\\{-1,0,1\\}\ \forall\lambda\in
E\right\\}\leq\left\langle E\right\rangle=\text{Sd}(\varphi_{E}).$ (4)

In particular, $\text{Sd}(\phi_{E})$ is still a finite set. For any
$m\in\mathbb{N}$ we have a state-preserving embedding

$(\mathbb{M}_{E}^{\otimes m},\phi_{E}^{\otimes
m})\hookrightarrow(R_{E},\varphi_{E}),$

given by mapping the $m$ tensor factors of
$(M_{2}(\mathbb{C}),\phi_{\lambda})$ into the first $m$ tensor factors of

$(R_{\lambda},\varphi_{\lambda})=\bigotimes_{n\in\mathbb{N}}(M_{2}(\mathbb{C}),\phi_{\lambda}).$

It is easy to see that this image is $\varphi_{E}$-invariant, and so there
exists a $\varphi_{E}$-preserving faithful normal conditional expectation
$\mathcal{E}_{E,m}\colon R_{E}\to\mathbb{M}_{E}^{\otimes m}$.

######  Theorem 3.2.

Let $N$ be a diffuse separable injective factor equipped with an extremal
almost periodic state $\varphi$. For every finite set $x_{1},\ldots,x_{n}\in
N$, finite set $E_{0}\subset\text{Sd}(\varphi)$, and $\epsilon>0$, there
exists a finite subset $E_{0}\subset E\subset\text{Sd}(\varphi)$,
$m\in\mathbb{N}$, and normal unital completely positive maps

$S\colon N\to\mathbb{M}_{E}^{\otimes m}\qquad\text{ and }\qquad
T\colon\mathbb{M}_{E}^{\otimes m}\to N,$

such that

$\displaystyle\phi_{E}^{\otimes m}\circ S$ $\displaystyle=\varphi,$
$\displaystyle\varphi\circ T$ $\displaystyle=\phi_{E}^{\otimes m}$
$\displaystyle\sigma_{t}^{\phi_{E}^{\otimes m}}\circ S$
$\displaystyle=S\circ\sigma_{t}^{\varphi},$
$\displaystyle\sigma_{t}^{\varphi}\circ T$
$\displaystyle=T\circ\sigma_{t}^{\phi_{E}^{\otimes m}}$

for all $t\in\mathbb{R}$, and

$\|T\circ S(x_{k})-x_{k}\|_{\varphi}<\epsilon\qquad k=1,\ldots,n$

###### Proof.

Apply Lemma 3.1 to $x_{1},\ldots,x_{n}$, $\frac{\epsilon}{2}$, and $E_{0}$ to
find normal unital completely positive maps $S^{\prime}\colon N\to R$ and
$T^{\prime}\colon R\to N$ and a faithful normal state $\psi$ on $R$ with their
suite of properties. Let $E:=\text{Sd}(\psi)$, which satisfies $E_{0}\subset
E\subset\text{Sd}(\varphi)$. Applying Lemma 2.4 to $(R_{E},\varphi_{E})$ and
$(R,\psi)$ (note that (4) gives $E\subset\text{Sd}(\varphi_{E})$), we can
identify $R$ as a $\varphi_{E}$-invariant subalgebra of $R_{E}$ satisfying
$\varphi_{E}|_{R}=\psi$. Consequently, there exists a faithful normal
$\varphi_{E}$-preserving conditional expectation $\mathcal{E}_{R}:R_{E}\to R$,
and so we define normal unital completely positive maps

$S^{\prime\prime}\colon N\to R_{E}\qquad\text{ and }\qquad
T^{\prime\prime}\colon R_{E}\to N,$

by letting $S^{\prime\prime}$ be $S^{\prime}$ but viewed as valued in $R_{E}$
and letting $T^{\prime\prime}:=T^{\prime}\circ\mathcal{E}_{R}$. It follows
that

$\displaystyle\varphi_{E}\circ S^{\prime\prime}$ $\displaystyle=\psi\circ
S^{\prime}=\varphi$ $\displaystyle\varphi\circ T^{\prime\prime}$
$\displaystyle=\varphi\circ
T^{\prime}\circ\mathcal{E}_{R}=\psi\circ\mathcal{E}_{R}=\varphi_{E}$

and

$\displaystyle\sigma_{t}^{\varphi_{E}}\circ S^{\prime\prime}$
$\displaystyle=\sigma_{t}^{\psi}\circ
S^{\prime}=S^{\prime}\circ\sigma_{t}^{\varphi}=S^{\prime\prime}\circ\sigma_{t}^{\varphi}$
$\displaystyle\sigma_{t}^{\varphi}\circ T^{\prime\prime}$
$\displaystyle=\sigma_{t}^{\varphi}\circ
T^{\prime}\circ\mathcal{E}_{R}=T^{\prime}\circ\sigma_{t}^{\psi}\circ\mathcal{E}_{R}=T^{\prime\prime}\circ\sigma_{t}^{\varphi_{E}},$

for all $t\in\mathbb{R}$. Moreover, for each $k=1,\ldots,n$ we have

$\|T^{\prime\prime}\circ
S^{\prime\prime}(x_{k}))-x_{k}\|_{\varphi}=\|T^{\prime}\circ\mathcal{E}_{R}\circ
S^{\prime}(x_{k})-x_{k}\|_{\varphi}=\|T^{\prime}\circ
S^{\prime}(x_{k})-x_{k}\|_{\varphi}<\frac{\epsilon}{2}.$

Now, as in the discussion preceding the statement of the theorem we view
$\mathbb{M}_{E}^{\otimes m}$ as a subfactor of $R_{E}$ with
$\varphi_{E}$-preserving conditional expectation $\mathcal{E}_{E,m}\colon
R_{E}\to\mathbb{M}_{E}^{\otimes m}$ for each $m\in\mathbb{N}$. Then for $x\in
R_{E}$ we have

$\lim_{m\to\infty}\|\mathcal{E}_{E,m}(x)-x\|_{\varphi_{E}}=0.$

Choose $m\in\mathbb{N}$ large enough so that

$\|\mathcal{E}_{E,m}\circ
S^{\prime\prime}(x_{k})-S^{\prime\prime}(x_{k})\|_{\varphi_{E}}<\frac{\epsilon}{2},$

for each $k=1,\ldots,n$. Finally, we define the desired normal unital
completely positive maps $S$ and $T$ by

$S:=\mathcal{E}_{E,m}\circ S^{\prime\prime}\qquad\text{ and }\qquad
T:=T^{\prime\prime}|_{\mathbb{M}_{E}^{\otimes m}}.$

The claimed relations with $\varphi$ and $\sigma_{t}^{\varphi}$,
$t\in\mathbb{R}$, are immediate, and for each $k=1,\ldots,n$ we have

$\displaystyle\|T\circ S(x_{k})-x_{k}\|_{\varphi}$ $\displaystyle\leq\|(T\circ
S-T^{\prime\prime}\circ
S^{\prime\prime})(x_{k})\|_{\varphi}+\|T^{\prime\prime}\circ
S^{\prime\prime}(x_{k})-x_{k}\|_{\varphi}$
$\displaystyle=\|T^{\prime\prime}\left[\mathcal{E}_{E,m}\circ
S^{\prime\prime}(x_{k})-S^{\prime\prime}(x_{k})\right]\|_{\varphi}+\|T^{\prime\prime}\circ
S^{\prime\prime}(x_{k})-x_{k}\|_{\varphi}<\frac{\epsilon}{2}+\frac{\epsilon}{2}=\epsilon,$

as desired. ∎

## 4\. Approximate Kraus representations

For $(N,\varphi)$ a diffuse factor equipped with an extremal almost periodic
state, the results in this section concern $\varphi$-invariant finite
dimensional subfactors $F\leq N$ and normal unital completely positive maps
$T\colon F\to N$ that are compatible with $\varphi$ and its modular theory. We
note that Theorem 3.2 yields such maps after using Lemma 2.5 to identify
$\mathbb{M}_{E}^{\otimes}$ with such a subfactor $F\leq N$.

Our first lemma, corresponds to [Haa89, Lemma 4.4], which shows such maps $T$
admit Kraus representations that use only elements of the centralizer
$N^{\varphi}$.

######  Lemma 4.1.

Let $N$ be a factor equipped with an extremal almost periodic state $\varphi$,
and let $F\leq N$ be a finite dimensional subfactor for which

$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}.$

If $T\colon F\to N$ is a normal completely positive map satisfying
$\sigma_{t}^{\varphi}\circ T=T\circ\sigma_{t}^{\varphi|_{F}}$, then there
exists a finite set $a_{1},\ldots,a_{n}\in N^{\varphi}$ such that

$T(x)=\sum_{j=1}^{n}a_{j}^{*}xa_{j}$

for all $x\in F$.

###### Proof.

We can find a family of matrix units $\\{e_{i,j}\in F\colon 1\leq i,j\leq
d\\}$ so that $\varphi|_{F}=\text{Tr}(h\,\cdot\,)$ and

$h=\sum_{i=1}^{d}\mu_{i}e_{i,i},$

where $\sum_{i}\mu_{i}=1$. Then $\\{\frac{\mu_{i}}{\mu_{j}}\colon 1\leq
i,j\leq d\\}=\text{Sd}(\varphi|_{F})\subset\text{Sd}(\varphi)$. The complete
positivity of $T$ implies

$a:=\sum_{i,j=1}^{d}T(e_{i,j})\otimes e_{i,j}$

is a positive operator in $N\otimes F$, so we can define $b:=a^{1/2}$. Then
$b$ is of the form

$b=\sum_{i,j=1}^{d}b_{i,j}\otimes e_{i,j}$

for $b_{i,j}\in N$ and

$T(e_{i,j})=(1\otimes\text{Tr})(a(1\otimes
e_{j,i}))=(1\otimes\text{Tr})(b^{*}b(1\otimes
e_{j,i}))=\sum_{k=1}^{n}b_{k,i}^{*}b_{k,j}.$

If we set $c_{i,j}:=\sum_{k=1}^{d}e_{k,j}b_{i,k}$ for each $1\leq i,j\leq d$,
then it follows that

$\sum_{i,j=1}^{d}c_{i,j}^{*}e_{k,\ell}c_{i,j}=\sum_{i,j,r,s=1}^{d}b_{i,r}^{*}e_{j,r}e_{k,\ell}e_{s,j}b_{i,s}=\sum_{i,j=1}^{d}b_{i,r}^{*}e_{j,j}b_{i,s}=T(e_{k,\ell}).$

Therefore

$T(x)=\sum_{i,j=1}^{d}c_{i,j}^{*}xc_{i,j}$

for all $x\in F$. Next we will replace the $c_{i,j}$ with elements of the
centralizer. Define a new state on $F$ by $\psi:=\text{Tr}(h^{-1}\,\cdot\,)$.
Then using $\sigma_{t}^{\varphi}\circ T=T\circ\sigma_{t}^{\varphi|_{F}}$ we
have

$\sigma_{t}^{\varphi\otimes\psi}(a)=\sum_{i,j=1}^{d}\sigma_{t}^{\varphi}(T(e_{i,j}))\otimes\sigma_{t}^{\psi}(e_{i,j})=\sum_{i,j=1}^{d}(\frac{\mu_{i}}{\mu_{j}})T(e_{i,j})\otimes\frac{\mu_{j}}{\mu_{i}}e_{i,j}=a.$

Hence $a\in(N\otimes F)^{\varphi\otimes\psi}$ and therefore $b$ lies in this
centralizer as well. Running a similar computation as above on the expanded
formula for $b$ yields
$\sigma_{t}^{\varphi}(b_{i,j})=\frac{\mu_{i}}{\mu_{j}}b_{i,j}$ for all
$i,j=1,\ldots,d$, and therefore

$\sigma_{t}^{\varphi}(c_{i,j})=\sum_{k=1}^{d}\frac{\mu_{k}}{\mu_{j}}e_{k,j}\frac{\mu_{i}}{\mu_{k}}b_{i,k}=\frac{\mu_{i}}{\mu_{j}}c_{i,j}$

for all $i,j=1,\ldots,d$.

Next, observe that $\varphi|_{F^{c}}$ is extremal almost periodic with
$\text{Sd}(\varphi|_{F})\subset\text{Sd}(\varphi|_{F^{c}})$ by Lemma 2.2. We
claim that for each $\frac{\mu_{i}}{\mu_{j}}\in\text{Sd}(\varphi|_{F})$ there
exists a finite set
$\mathcal{V}_{i,j}\subset(F^{c})^{(\varphi|_{F^{c}},\frac{\mu_{i}}{\mu_{j}})}$
satisfying

$\sum_{v\in\mathcal{V}_{i,j}}vv^{*}=1.$

Indeed, we first use [GLN24, Lemma 2.1] to a find a (potentially infinite)
family
$\mathcal{W}\subset(F^{c})^{(\varphi|_{F^{c}},\frac{\mu_{i}}{\mu_{j}})}$
satisfying

$\sum_{v\in\mathcal{W}}vv^{*}=1.$

For $\frac{\mu_{i}}{\mu_{j}}\geq 1$, this same lemma implies we can chose
$\mathcal{V}_{i,j}=\mathcal{W}$ of size one. Thus it suffices to consider the
case $\frac{\mu_{i}}{\mu_{j}}<1$. For some $m\in\mathbb{N}$ we can partition
$\mathcal{W}=\mathcal{W}_{1}\sqcup\cdots\sqcup\mathcal{W}_{m}$ so that

$\sum_{v\in\mathcal{W}_{k}}\varphi(vv^{*})\leq\frac{\mu_{i}}{\mu_{j}}$

for each $k=1,\ldots,m$. Consequently,

$\sum_{v\in\mathcal{W}_{k}}\varphi(v^{*}v)=\sum_{v\in\mathcal{W}_{k}}\frac{\mu_{j}}{\mu_{i}}\varphi(vv^{*})\leq
1,$

and so by precomposing the $v\in\mathcal{W}_{k}$ with suitable partial
isometries in $(F^{c})^{\varphi|_{F^{c}}}$ we can assume the range and source
projections are pairwise orthogonal over $\mathcal{W}_{k}$ (this uses the
factoriality of $(F^{c})^{\varphi|_{F^{c}}}$). Setting
$v_{k}:=\sum_{v\in\mathcal{W}_{k}}v$ for each $k=1,\ldots,m$, the desired
finite set is then $\mathcal{V}_{i,j}:=\\{v_{1},\ldots,v_{m}\\}$.

Now, $v^{*}c_{i,j}\in N^{\varphi}$ for each $1\leq i,j\leq d$ and
$v\in\mathcal{V}_{i,j}$, and since $F^{c}$ commutes with $F$ inside $N$ we
have

$\sum_{i,j=1}^{d}\sum_{v\in\mathcal{V}_{i,j}}(v^{*}c_{i,j})^{*}x(v^{*}c_{i,j})=\sum_{i,j=1}^{d}\sum_{v\in\mathcal{V}_{i,j}}c_{i,j}^{*}xvv^{*}c_{i,j}=\sum_{i,j=1}^{d}c_{i,j}^{*}xc_{i,j}=T(x)$

for each $x\in F$. Thus $\\{v^{*}c_{i,j}\colon 1\leq i,j\leq d,\
v\in\mathcal{V}_{i,j}\\}$ is the desired finite set in $N^{\varphi}$. ∎

The next pair of lemmas are relative Dixmier norm approximation results. These
relate to [Haa89, Corollary 3.4 and Lemma 4.5], respectively, but are
necessarily weaker in the case that $\text{Sd}(\varphi)$ is dense in
$\mathbb{R}_{+}$ (see the proof of [Haa89, Theorem 3.1]).

######  Lemma 4.2.

Let $N$ be a diffuse factor with an extremal almost periodic state $\varphi$.
For any finite set $x_{1},\ldots,x_{n}\in N^{(\varphi,\text{eig})}$ and
$\epsilon>0$ there exists
$\alpha\in\operatorname{conv}\\{\text{Ad}\left(u\right)\colon u\in
N^{\varphi}\\}$ satisfying

$\|\alpha(x_{k})-\varphi(x_{k})\|<\epsilon$

for each $k=1,\ldots,n$.

###### Proof.

We first observe that for $\lambda\in\mathbb{R}_{+}$ and $x\in
N^{(\varphi,\lambda)}$ one has

$\varphi(x)\in\overline{\operatorname{conv}\\{u^{*}xu\,|\,u\in
U(N_{\varphi)}\\}}^{\|\cdot\|}.$

Indeed, for $\lambda=1$ this is simply the Dixmier approximation theorem for
type $\mathrm{II}_{1}$ factors (see [Dix81, Section III.5.1]), and for
$\lambda\neq 1$ this follows from the same argument as in the proof of [Haa89,
Lemma 3.2]. (Note that $N^{\varphi}$ is indeed a type $\mathrm{II}_{1}$ factor
by [GLN24, Lemma 3.1].)

Now, let $E\subset\text{Sd}(\varphi)$ be the necessarily finite set such that
$x_{1},\ldots,x_{n}\in\text{span}\\{N^{(\varphi,\lambda)}\colon\lambda\in
E\\}$, and for each $k=1,\ldots,n$ write $x_{k}=\sum_{\lambda\in
E}x_{k,\lambda}$ where $x_{k,\lambda}\in N^{(\varphi,\lambda)}\cup\\{0\\}$.
Iteratively applying our initial observation through some enumeration of
$\\{1,\ldots,n\\}\times E$ yields
$\alpha\in\operatorname{conv}\\{\text{Ad}\left(u\right)\colon u\in
N^{\varphi}\\}$ satisfying

$\|\alpha(x_{k,\lambda})-\varphi(x_{k,\lambda})\|<\frac{\epsilon}{|E|}$

for each $k=1,\ldots,n$ and $\lambda\in E$, and so the claimed inequality for
$x_{k}$ follows. ∎

######  Lemma 4.3.

Let $N$ be a diffuse factor equipped with an extremal almost periodic state
$\varphi$, and let $F\leq N$ be a finite dimensional subfactor for which

$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}.$

If $\mathcal{E}_{F}\colon N\to F$ is the $\varphi$-invariant conditional
expectation, then for all $x\in N^{(\varphi,\text{eig})}$ and $\epsilon>0$
there exists $\alpha\in\operatorname{conv}\\{\text{Ad}\left(u\right)\colon
u\in\mathcal{U}(F^{c}\cap N^{\varphi})\\}$ satisfying

$\|\alpha(x)-\mathcal{E}_{F}(x)\|<\epsilon.$

###### Proof.

For $x\in N^{(\varphi,\text{eig})}$ write

$x=\sum_{\lambda\in E}x_{\lambda},$

where $x_{\lambda}\in N^{(\varphi,\lambda)}$ for each $\lambda$ in a finite
subset $E\subset\text{Sd}(\varphi)$. Fix a family of matrix units
$\\{e_{i,j}\colon 1\leq i,j\leq d\\}$ for $F$ consisting of eigenoperators for
$\sigma^{\varphi|_{F}}$. In fact,
$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}$ implies they will also be
eigenoperators for $\sigma^{\varphi}$, say
$\sigma_{t}^{\varphi}(e_{i,j})=\lambda_{i,j}^{it}e_{i,j}$ for
$\lambda_{i,j}\in\text{Sd}(\varphi)$. For each $\lambda\in E$ write

$x_{\lambda}=\sum_{i,j=1}^{d}e_{i,j}x_{i,j}^{(\lambda)},$

where $x_{i,j}^{(\lambda)}\in F^{c}$ for all $1\leq i,j\leq d$. Our choice of
matrix units and $x_{\lambda}\in N^{(\varphi,\lambda)}$ implies
$x_{i,j}^{(\lambda)}\in N^{(\varphi,\frac{\lambda}{\lambda_{i,j}})}$ for each
$1\leq i,j\leq d$. Also, the tensor decomposition of $\varphi$ implies

$\mathcal{E}_{F}(x_{\lambda})=\sum_{i,j=1}^{d}e_{i,j}\varphi(x_{i,j}^{(\lambda)}).$

Lemma 2.2 tells us that $\varphi|_{F^{c}}$ is extremal almost periodic, and so
we can apply Lemma 4.2 to $\\{x_{i,j}^{(\lambda)}\colon\lambda\in E,\ 1\leq
i,j\leq d\\}\subset F^{c}$ to find
$\alpha\in\operatorname{conv}\\{\text{Ad}\left(u\right)\colon
u\in\mathcal{U}(F^{c}\cap N^{\varphi})\\}$ satisfying

$\|\alpha(x_{i,j}^{(\lambda)})-\varphi(x_{i,j}^{(\lambda)})\|<\frac{\epsilon}{d^{2}|E|}.$

(Note that $F^{c}\cap N^{\varphi}=(F^{c})^{\varphi|_{F^{c}}}$.) Since $F^{c}$
commutes with $F$ we have

$\|\alpha(x)-\mathcal{E}_{F}(x)\|=\left\|\sum_{\lambda\in
E}\sum_{i,j=1}^{d}e_{i,j}\left[\alpha(x_{i,j}^{(\lambda)})-\varphi(x_{i,j}^{(\lambda)})\right]\right\|<\epsilon,$

as claimed. ∎

The following theorem is our main result in this section and corresponds to
[Haa89, Theorem 4.1]. Recall that Lemma 4.1 gave an exact Kraus representation
of the completely positive map $T(x)=\sum_{j=1}^{n}a_{j}^{*}xa_{j}$ for
$a_{1},\ldots,a_{n}\in N^{\varphi}$. Since $T$ is unital, we in particular
have that $\sum_{j=1}^{n}a_{j}^{*}a_{j}=1$. However,
$\sum_{j=1}^{n}a_{j}a_{j}^{*}=1$ may fail to hold, and it will be needed in
the proof of Theorem A to apply [Haa89, Theorem 2.3]. In the following
theorem, we obtain this latter relation by sacrificing our exact finite Kraus
representation for an infinite approximate Kraus representation.

######  Theorem 4.4.

Let $N$ be a diffuse factor equipped with an extremal almost periodic state
$\varphi$, and let $F\leq N$ be a finite dimensional subfactor for which

$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}$

Suppose $T\colon F\to N$ is a normal unital completely positive map satisfying
$\varphi\circ T=\varphi|_{F}$ and $\sigma_{t}^{\varphi}\circ
T=T\circ\sigma_{t}^{\varphi|_{F}}$. Then for every $\delta>0$ there exists a
sequence $(a_{n})_{n\in\mathbb{N}}\subset N^{\varphi}$ such that

$\sum_{n=1}^{\infty}a_{n}^{*}a_{n}=\sum_{n=1}^{\infty}a_{n}a_{n}^{*}=1$

and

$\left\|T(x)-\sum_{n=1}^{\infty}a_{n}^{*}xa_{n}\right\|\leq\delta\|x\|$

for all $x\in F$.

###### Proof.

We first apply Lemma 4.1 to obtain $b_{1},\ldots,b_{n}\in N^{\varphi}$ so that

$T(x)=\sum_{i=1}^{n}b_{i}^{*}xb_{i},$

for all $x\in F$. In particular, this gives

$\sum_{i=1}^{n}b_{i}^{*}b_{i}=T(1)=1.$

Let $\mathcal{E}_{F}\colon N\to F$ to be $\sigma^{\varphi}$-preserving
conditional expectation. Using $\varphi\circ T=\varphi|_{F}$ and $b_{i}\in
N^{\varphi}$, we have for $x\in F$

$\left\langle\mathcal{E}_{F}\left(\sum_{i=1}^{d}b_{i}b_{i}^{*}\right),x\right\rangle_{\varphi}=\left\langle\sum_{i=1}^{d}b_{i}b_{i}^{*},x\right\rangle_{\varphi}=\left\langle
1,\sum_{i=1}^{d}b_{i}^{*}xb_{i}\right\rangle_{\varphi}=\left\langle
1,T(x)\right\rangle_{\varphi}=\left\langle 1,x\right\rangle_{\varphi}.$

Hence $\mathcal{E}_{F}(\sum_{i=1}^{d}b_{i}b_{i}^{*})=1$.

Now, apply Lemma 4.3 to $\sum_{i=1}^{d}b_{i}b_{i}^{*}$ to find
$\alpha=\sum_{j=1}^{d}t_{j}\text{Ad}\left(u_{j}\right)\in\operatorname{conv}\\{\text{Ad}\left(u\right)\colon
u\in\mathcal{U}(F^{c}\cap N^{\varphi})\\}$ satisfying

$\left\|\alpha\left(\sum_{i=1}^{n}b_{i}b_{i}^{*}\right)-1\right\|<\frac{\delta}{2}.$

Set $c_{i,j}:=t_{j}^{\frac{1}{2}}u_{j}b_{i}$. Then the above inequality gives

$\displaystyle\|\sum_{i=1}^{n}\sum_{j=1}^{d}c_{i,j}c_{i,j}^{*}-1\|<\frac{\delta}{2}.$
(5)

We also have

$\sum_{i=1}^{n}\sum_{j=1}^{d}c_{i,j}^{*}c_{i,j}=\sum_{j=1}^{d}t_{j}\sum_{i=1}^{n}b_{i}^{*}b_{i}=1,$

and for $x\in F$ we have

$\sum_{i=1}^{n}\sum_{j=1}^{d}c_{i,j}^{*}xc_{i,j}=\sum_{j=1}^{d}t_{j}\sum_{i=1}^{n}b_{i}^{*}xb_{i}=T(x).$

Reindex the operators $c_{i,j}$ as $c_{1},\ldots,c_{m}$ for some
$m\in\mathbb{N}$, and then set

$a_{i}:=\left(1-\frac{\delta}{2}\right)^{\frac{1}{2}}c_{i}.$

for each $i=1,\ldots,m$. Then

$\sum_{i=1}^{m}a_{i}^{*}a_{i}=1-\frac{\delta}{2}<1,$

and (5) implies

$\sum_{i=1}^{m}a_{i}a_{i}^{*}=\left[\sum_{i=1}^{m}a_{i}a_{i}^{*}-\left(1-\frac{\delta}{2}\right)\right]+\left(1-\frac{\delta}{2}\right)<\left(1-\frac{\delta}{2}\right)\left(\frac{\delta}{2}\left(1-\frac{\delta}{2}\right)+1\right)<1.$

Recalling that $N^{\varphi}$ is a type $\mathrm{II}_{1}$ factor by [GLN24,
Lemma 3.1], we can use [Haa85, Lemma 5.1] to find $\\{a_{i}\in
N^{\varphi}\colon i\geq m+1\\}$ so that

$\sum_{i=1}^{\infty}a_{i}^{*}a_{i}=\sum_{i=1}^{\infty}a_{i}a_{i}^{*}=1.$

Note that $\sum_{i\geq m+1}a_{i}^{*}a_{i}=\frac{\delta}{2}$, and consequently

$S(x):=\sum_{i=m+1}^{\infty}a_{i}^{*}xa_{i}$

is a completely positive map with $\|S\|=\|S(1)\|=\frac{\delta}{2}$. Thus
$\|S(x)\|\leq\frac{\delta}{2}\|x\|$, and recall that $\|T(x)\|\leq\|x\|$ since
$\|T\|=\|T(1)\|=1$. Hence

$\left\|T(x)-\sum_{i=1}^{\infty}a_{i}^{*}xa_{i}\right\|=\left\|T(x)-\left(1-\frac{\delta}{2}\right)T(x)-S(x)\right\|=\left\|\frac{\delta}{2}T(x)-S(x)\right\|\leq\delta\|x\|$

for all $x\in F$. ∎

## 5\. Proof of Theorem A

Let $N$ be a separable injective factor equipped with an extremal almost
periodic state $\varphi$. For $\Lambda:=\text{Sd}(\varphi)$, let
$(R_{\Lambda},\varphi_{\Lambda})$ be as in (2). Towards constructing the
isomorphism $(N,\varphi)\cong(R_{\Lambda},\varphi_{\Lambda})$, we first prove
three claims corresponding to [Haa89, Lemmas 6.2, 6.3, and 6.4], respectively.

Claim 1: For each finite set $u_{1},\ldots,u_{n}\in\mathcal{U}(N)$, finite set
$E_{0}\subset\text{Sd}(\varphi)$, and $\delta>0$ there exists a finite set
$E_{0}\subset E\subset\text{Sd}(\varphi)$, $m\in\mathbb{N}$, a completely
positive map $T\colon\mathbb{M}_{E}^{\otimes m}\to N$, and unitarty operators
$v_{1},\ldots,v_{n}\in\mathbb{M}_{E}^{\otimes m}$ such that

$\displaystyle\varphi\circ T$ $\displaystyle=\phi_{E}^{\otimes m}$
$\displaystyle\sigma_{t}^{\varphi}\circ T$
$\displaystyle=T\circ\sigma_{t}^{\phi_{E}^{\otimes m}}$

for all $t\in\mathbb{R}$, and

$\displaystyle\|T(v_{k})-u_{k}\|_{\varphi}<\delta$

for each $k=1,\ldots,n$.

Choose $\epsilon>0$ satisfying $(2\epsilon)^{\frac{1}{2}}+\epsilon<\delta$.
Applying Theorem 3.2 to $u_{1},\ldots,u_{n}\in N$, $E_{0}$, and this
$\epsilon$ we obtain a finite set $E_{0}\subset E\subset\text{Sd}(\varphi)$,
$m\in\mathbb{N}$, and normal unital completely positive maps $S\colon
N\to\mathbb{M}_{E}^{\otimes m}$ and $T\colon\mathbb{M}_{E}^{\otimes m}\to N$
with their collection of properties. Set $y_{k}:=S(u_{k})$ so that
$\|y_{k}\|\leq 1$ and

$\|T(y_{k})-u_{k}\|_{\varphi}<\epsilon$

for each $k=1,\ldots,n$. Observe that

$\|y_{k}\|_{\phi_{E}^{\otimes
m}}\geq\|T(y_{k})\|_{\varphi}\geq\|u_{k}\|_{\varphi}-\|T(y_{k})-u_{k}\|_{\varphi}>1-\epsilon.$

Let $y_{k}=v_{k}|y_{k}|$ be the polar decomposition, and since
$\mathbb{M}_{E}^{\otimes m}$ is a finite factor we can extend $v_{k}$ to a
unitary in this algebra. Thus

$\|v_{k}-y_{k}\|_{\phi_{E}^{\otimes m}}^{2}=\|1-|y_{k}|\|_{\phi_{E}^{\otimes
m}}^{2}\leq 1-\||y_{k}|\|_{\phi_{E}^{\otimes
m}}^{2}=1-\|y_{k}\|_{\phi_{E}^{\otimes m}}^{2}<1-(1-\epsilon)^{2}<2\epsilon,$

where in the first inequality we have used $|y_{k}|^{2}+(1-|y_{k}|)^{2}\leq
1$, which follows from $0\leq|y_{k}|\leq\|y_{k}\|\leq 1.$ Consequently we have

$\|T(v_{k})-u_{k}\|_{\varphi}\leq\|T(v_{k}-y_{k})\|_{\varphi}+\|T(y_{k})-u_{k}\|_{\varphi}<(2\epsilon)^{\frac{1}{2}}+\epsilon<\delta,$

by our original assumption on $\epsilon$. This proves Claim 1.

Claim 2: For each finite set $u_{1},\ldots,u_{n}\in\mathcal{U}(N)$, finite set
$E_{0}\subset\text{Sd}(\varphi)$, and $\delta>0$ there exists a finite
dimensional subfactor $F\leq N$, unitary operators
$v_{1},\ldots,v_{n}\in\mathcal{U}(F)$, and a sequence
$(a_{i})_{i\in\mathbb{N}}\subset N^{\varphi}$ satisfying

  1. (i)

$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}$;

  2. (ii)

$(F,\varphi|_{F})\cong(\mathbb{M}_{E}^{\otimes m},\phi_{E}^{\otimes m})$ for
some finite set $E_{0}\subset E\subset\text{Sd}(\varphi)$ and
$m\in\mathbb{N}$;

  3. (iii)

$\sum_{i=1}^{\infty}a_{i}^{*}a_{i}=\sum_{i=1}^{\infty}a_{i}a_{i}^{*}=1$;

  4. (iv)

$\sum_{i=1}^{\infty}\|v_{k}a_{i}-a_{i}u_{k}\|_{\varphi}^{2}<\delta$ for each
$k=1,\ldots,n$.

First, we apply Claim 1 to $u_{1},\ldots,u_{n}$, $E_{0}$, and
$\frac{\delta}{4}$ and let $E_{0}\subset E\subset\text{Sd}(\varphi)$,
$m\in\mathbb{N}$, $T\colon\mathbb{M}_{E}^{\otimes m}\to N$, and
$v_{1},\ldots,v_{n}\in\mathbb{M}_{E}^{\otimes m}$ the resulting data. Then
applying Lemma 2.5 to $(\mathbb{M}_{E}^{\otimes m},\phi_{E}^{\otimes m})$
gives a finite dimensional subfactor $F\leq N$ satisfying (i) and (ii).

Observe that identifying $(\mathbb{M}_{E}^{\otimes m},\phi_{E}^{\otimes m})$
with $(F,\varphi|_{F})$ gives $\varphi\circ T=\varphi|_{F}$ and
$\sigma_{t}^{\varphi}\circ T=T\circ\sigma_{t}^{\varphi|_{F}}$ for all
$t\in\mathbb{R}$. Thus we can apply Theorem 4.4 to $F$, $T$, and
$\frac{\delta}{4}$ to obtain $(a_{i})_{i\in\mathbb{N}}\subset N^{\varphi}$
satisfying (iii) in Claim 2 and

$\left\|T(x)-\sum_{i=1}^{\infty}a_{i}^{*}xa_{i}\right\|<\frac{\delta}{4}\|x\|$

for all $x\in F$. Using this as well as the proximity of $T(v_{k})$ and
$u_{k}$ given by Claim 1 we have

$\left\|u_{k}-\sum_{i=1}^{\infty}a_{i}^{*}v_{k}a_{i}\right\|_{\varphi}\leq\|u_{k}-T(v_{k})\|_{\varphi}+\left\|T(v_{k})-\sum_{i=1}^{\infty}a_{i}^{*}v_{k}a_{i}\right\|_{\varphi}<\frac{\delta}{2}.$

We also compute

$\sum_{i=1}^{\infty}\|a_{i}u_{k}\|_{\varphi}^{2}=\sum_{i=1}^{\infty}\varphi(u_{k}^{*}a_{i}^{*}a_{i}u_{k})=\varphi(u_{k}^{*}u_{k})=1,$

and

$\sum_{i=1}^{\infty}\|v_{k}a_{i}\|_{\varphi}^{2}=\sum_{i=1}^{\infty}\varphi(a_{i}^{*}a_{i})=1.$

Putting all of this together we obtain

$\displaystyle\sum_{i=1}^{\infty}\|v_{k}a_{i}-a_{k}u_{k}\|_{\varphi}^{2}$
$\displaystyle=2-2\sum_{i=1}^{\infty}\text{Re}\
\varphi(u_{k}^{*}a_{i}^{*}v_{k}a_{i})$ $\displaystyle=2\text{Re}\
\varphi\left(u_{k}^{*}\left(u_{k}-\sum_{i=1}^{\infty}a_{i}^{*}v_{k}a_{i}\right)\right)\leq
2\left\|u_{k}-\sum_{i=1}^{\infty}a_{i}^{*}v_{k}a_{i}\right\|_{\varphi}<\delta.$

This proves (iv) and hence all of Claim 2.

Claim 3: For each finite set $x_{1},\ldots,x_{n}\in N$, finite set
$E_{0}\subset\text{Sd}(\varphi)$, and $\epsilon>0$ there exists a finite
dimensional subfactor $F\leq N$ and a finite set $y_{1},\ldots,y_{n}\in F$
satisfying:

  1. (i)

$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}$;

  2. (ii)

$(F,\varphi|_{F})\cong(\mathbb{M}_{E}^{\otimes m},\phi_{E}^{\otimes m})$ for
some finite set $E_{0}\subset E\subset\text{Sd}(\varphi)$ and
$m\in\mathbb{N}$;

  3. (iii)

$\|x_{k}-y_{k}\|_{\varphi}<\epsilon$ for each $k=1,\ldots,n$.

Decomposing the $x_{i}$ into a linear combintation of four unitaries, it
suffices to consider $u_{1},\ldots,u_{n}\in\mathcal{U}(N)$. Apply Claim 2 to
$u_{1},\ldots,u_{n}$, $E_{0}$, and some $\delta>0$ to be determined later, and
let $G\leq N$ and $(a_{i})_{i\in\mathbb{N}}\subset N^{\varphi}$ be the
resulting data. The $N$-$N$-bimodule structure of $L^{2}(N,\varphi)$ restricts
to an $N^{\varphi}$-$N^{\varphi}$-bimodule structure satisfying

$a\cdot x\cdot b=aJ_{\varphi}b^{*}J_{\varphi}x=axb$

for $a,b\in N^{\varphi}$ and $x\in N$. (Indeed, first check this for $x\in
N^{(\varphi,\text{eig})}$ for which
$J_{\varphi}x=\sigma_{-i/2}^{\varphi}(x^{*})$.) Claim 2 therefore tells us
that the tuples $(v_{1},\ldots,v_{n}),(u_{1},\ldots,u_{n})\in
L^{2}(N,\varphi)^{\oplus n}$ are $\delta$-related over $N^{\varphi}$ in the
sense of [Haa89, Definition 2.1]. So given $\epsilon>0$, if we take $\delta>0$
sufficiently small then [Haa89, Theorem 2.3] yields a unitary
$w\in\mathcal{U}(N^{\varphi})$ satisfying

$\|u_{k}-w^{*}v_{k}w\|_{\varphi}=\|wu_{k}-v_{k}w\|_{\varphi}<\epsilon$

for each $k=1,\ldots,n$. Define $F:=w^{*}Gw$ and $y_{k}=w^{*}v_{k}w$ for each
$k=1,\ldots,n$ so that (iii) holds. Since $w$ is in the centralizer for
$\varphi$, we have

$\varphi|_{F}\otimes\varphi|_{F^{c}}=(\varphi|_{G}\otimes\varphi|_{G^{c}})\circ(\text{Ad}\left(w^{*}\right)\otimes\text{Ad}\left(w^{*}\right))=\varphi|_{G}\otimes\varphi|_{G^{c}}=\varphi,$

so that (i) holds. Also

$(F,\varphi|_{F})\cong(G,\varphi|_{G})\cong(\mathbb{M}_{E}^{\otimes
m},\phi_{E}^{\otimes m}),$

for some finite set $E_{0}\subset E\subset\text{Sd}(\varphi)$ and
$m\in\mathbb{N}$ so that (ii) and hence Claim 3 holds.

We now begin the proof of our main theorem in earnest. Using separability of
$N$ we write a countable generating set $\\{x_{n}\in N\colon
n\in\mathbb{N}\\}$ for $N$. Also enumerate
$\text{Sd}(\varphi)\cap(0,1)=\\{\lambda_{n}\colon n\in\mathbb{N}\\}$. We will
inductively construct a sequence of commuting finite dimensional subfactors
$(F_{n})_{n\in\mathbb{N}}$ of $N$ and a sequence of finite subsets
$(E_{n})_{n\in\mathbb{N}}$ of $\text{Sd}(\varphi)$ satisfying:

  1. (a)

$\inf\\{\|x_{k}-y\|_{\varphi}\colon y\in F_{1}\vee\cdots\vee
F_{n}\\}<\frac{1}{n}$ for each $n\in\mathbb{N}$ and each $k=1,\ldots,n$;

  2. (b)

$\varphi=\varphi|_{F_{n}}\otimes\varphi|_{F_{n}^{c}}$ for each
$n\in\mathbb{N}$;

  3. (c)

$\\{\lambda_{1},\ldots,\lambda_{n}\\}\subset E_{n}\subset E_{n+1}$;

  4. (d)

for each $n\in\mathbb{N}$ there exists $m_{n}\in\mathbb{N}$ so that

$(F_{n},\varphi|_{F_{n}})\cong(\mathbb{M}_{E_{n}}^{\otimes
m_{n}},\phi_{E_{n}}^{\otimes m_{n}}).$

$F_{1}$ and $E_{1}$ are obtained by applying Claim 3 to $x_{1}$,
$E_{0}=\\{1\\}$, and $\epsilon=1$. Suppose we have constructed
$F_{1},\ldots,F_{n}$ as above. Set

$F:=F_{1}\vee\cdots\vee F_{n}=\bigotimes_{k=1}^{n}F_{k}.$

Then $F$ is $\sigma^{\varphi}$-invariant since (b) implies this is true for
each $F_{1},\ldots,F_{n}$, and hence
$\varphi=\varphi|_{F}\otimes\varphi|_{F^{c}}$. Fix a family of matrix units
$\\{e_{i,j}\in F\colon 1\leq i,j\leq d\\}$ so that for each $k=1,\ldots,n+1$

$x_{k}=\sum_{i,j=1}^{d}e_{i,j}x^{(k)}_{i,j}$

where $x^{(k)}_{i,j}\in F^{c}$. Set

$K:=\sum_{i,j=1}^{d}\|e_{i,j}\|_{\varphi}.$

Recall that $\varphi|_{F^{c}}$ is extremal almost periodic with
$\text{Sd}(\varphi|_{F^{c}})=\text{Sd}(\varphi)$ by Lemma 2.2, so we can apply
Claim 3 to $(F^{c},\varphi|_{F^{c}})$, the finite set $\\{x^{(k)}_{i,j}\colon
1\leq k\leq n+1,\ 1\leq i,j\leq d\\}$, the finite set $\\{\lambda_{n+1}\\}\cup
E_{n}$, and $\epsilon=\frac{1}{(n+1)K}$ to obtain: a finite subset
$E_{n+1}\subset\text{Sd}(\varphi)$ satisfying (c); a finite dimensional
subfactor $F_{n+1}\leq F^{c}$ satisfying
$\varphi|_{F^{c}}=\varphi|_{F_{n+1}}\otimes\varphi|_{F_{n+1}^{\prime}\cap
F^{c}}$ as well as (d); and a finite set $\\{y^{(k)}_{i,j}\in F_{n+1}\colon
1\leq k\leq n+1\,1\leq i,j\leq d\\}$ satisfying

$\|x_{i,j}^{(k)}-y_{i,j}^{(k)}\|_{\varphi}<\frac{1}{(n+1)K^{1/2}}$

for each $k=1,\ldots,n+1$ and $1\leq i,j\leq d$. Note that $F_{n+1}$ being a
subfactor of $F^{c}$ implies it commutes with $F_{1},\ldots,F_{n}$. Composing
the expectations from $N$ to $F^{c}$ and from $F^{c}$ to $F_{n+1}$ gives (b).
Towards establishing (a), set

$y_{k}:=\sum_{i,j=1}^{d}e_{i,j}y_{i,j}^{(k)}$

for each $k=1,\ldots,n+1$ so that $y_{k}\in F\vee F_{n+1}=F_{1}\vee\cdots\vee
F_{n+1}$ and using (b) we have

$\displaystyle\|x_{k}-y_{k}\|_{\varphi}^{2}$
$\displaystyle=\varphi\left(\sum_{i,j=1}^{d}e_{i,j}\sum_{r=1^{d}}(x_{\ell,i}^{(k)}-y_{\ell,i}^{(k)})^{*}(x_{\ell,j}^{(k)}-y_{\ell,j}^{(k)})\right)$
$\displaystyle=\sum_{i,j,r=1}^{d}\varphi(e_{i,j})\varphi((x_{\ell,i}^{(k)}-y_{\ell,i}^{(k)})^{*}(x_{\ell,j}^{(k)}-y_{\ell,j}^{(k)}))$
$\displaystyle\leq\sum_{i,j=1}\|e_{i,j}\|_{\varphi}\frac{1}{(n+1)^{2}K}\leq\frac{1}{(n+1)^{2}},$

and so (a) follows. Induction therefore gives us the claimed sequence
$(F_{n})_{n\in\mathbb{N}}$ of commuting subfactors $N$ and the sequence
$(E_{n})_{n\in\mathbb{N}}$ of finite subsets of $\text{Sd}(\varphi)$.

Now, if $\mathcal{E}_{n}$ is the $\varphi$=preserving conditional expectation
onto $F_{1}\vee\cdots\vee F_{n}$ then for each $k\leq n$ we have

$\|x_{k}-\mathcal{E}_{n}(x_{k})\|_{\varphi}\leq\frac{1}{n}$

by (a). That is, the bounded sequence
$(\mathcal{E}_{n}(x_{k}))_{n\in\mathbb{N}}$ converges to $x_{k}$ in
$\|\,\cdot\,\|_{\varphi}$-norm and therefore in the strong operator topology.
Consequently,

$\bigvee_{n\in\mathbb{N}}F_{n}=\left(\bigcup_{n\in\mathbb{N}}F_{n}\right)^{\prime\prime}$

contains $x_{k}$ for all $k\in\mathbb{N}$. Since $\\{x_{k}\colon
k\in\mathbb{N}\\}$ is a generating set for $N$, we see that the above must in
fact equal $N$. Further, (b) implies $\varphi$ restricted to
$F_{1}\vee\cdots\vee F_{n}$ conincides with

$\bigotimes_{k=1}^{n}\varphi|_{F_{k}}$

for each $n\in\mathbb{N}$. Consequently, (c) implies

$(N,\varphi)\cong\bigotimes_{n\in\mathbb{N}}(F_{n},\varphi|_{F_{n}})\cong\bigotimes_{n\in\mathbb{N}}(\mathbb{M}_{E_{n}}^{\otimes
m_{n}},\phi_{E_{n}}^{\otimes m_{n}}).$

The inclusions $\\{\lambda_{1},\ldots,\lambda_{n+1}\\}\subset E_{n}\subset
E_{n+1}$ implies that $(M_{2}(\mathbb{C}),\phi_{\lambda_{n}})$ appears
infinite often in the above tensor product for each $n\in\mathbb{N}$. Thus

$(N,\varphi)\cong\bigotimes_{\Lambda\cap(0,1)}(R_{\lambda},\varphi_{\lambda})=(R_{\Lambda},\varphi_{\Lambda}),$

as claimed.$\hfill\square$

## 6\. Proof of Theorem C

Using either [Izu93, Theorem 4.7] when $J$ is finite or [Tho19, Example 1]
when $J$ is infinte, it follows that
$\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime}$ is a separable injective type
$\mathrm{III}$ factor. Thus by Theorem A it suffices to show $\varphi$ is
extremal almost periodic with $\text{Sd}(\varphi)=\Lambda$. The generators of
$\mathcal{O}_{J}$ being eigenoperators with respect to $\varphi$ implies the
almost periodicity of $\varphi$, and moreover that
$\text{Sd}(\varphi)\subset\Lambda$. Since $\text{Sd}(\varphi)$ contains
generators of the group $\Lambda$, we will obtain $\text{Sd}(\varphi)=\Lambda$
once we know the former is a group. Therefore the proof will be complete once
we have argued that $\varphi$ is extremal, and this is essentially already
contained in the proof of [Tho19, Theorem 5].

Indeed, [Tho19, Theorem 5] asserts that the _Connes spectrum_ (see [Tak03,
Section XI.2]) of the modular automorphism group of $\sigma^{\varphi}$ is
given by

$\Gamma(\sigma^{\varphi})=\overline{\Lambda}.$

In fact, for each projection $q$ in the center of
$(\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime})^{\varphi}$ and each $j\in J$,
in the proof of [Tho19, Theorem 5] Thomsen exhibits a partial isometry
$w\in(\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime})^{(\varphi,\mu(j))}$ with
$qwq\neq 0$. It follows that

$\mu(J)\subset\text{Sd}(\varphi|_{q\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime}q})$

for all such $q$. Equip $\Lambda$ with the discrete topology, let $G$ be its
compact dual, and
$G\overset{\alpha}{\curvearrowright}\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime}$
be the point modular extension of the modular automorphism group of $\varphi$.
Then the above, in conjunction with [Con74, Equation (1)], shows that

$\mu(J)\subset\text{Sp}(\alpha^{(q)}),$

where $\alpha^{(q)}$ is the restriction of $\alpha$ to $qMq$ and
$\text{Sp}(\alpha^{(q)})$ is its _Arveson spectrum_ (see [Tak03, Section
XI.1]). Using [Con73, Proposition 2.2.2] we see that

$\mu(J)\subset\bigcap_{q}\text{Sp}(\alpha^{(q)})=\Gamma(\alpha).$

But since the Connes spectrum of $\alpha$ is a subgroup of $\Lambda$ it
follows that $\Gamma(\alpha)=\Lambda$. This in turn implies
$(\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime})^{\alpha}=(\pi_{\varphi}(\mathcal{O}_{J})^{\prime\prime})^{\varphi}$
is a factor by [Con73, Theorem 2.4.1]. That is, $\varphi$ is extremal.
$\hfill\square$

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