# The possible temperatures for flows on a simple AF algebra

Klaus Thomsen<EMAIL_ADDRESS>Department of Mathematics, Aarhus University,
Ny Munkegade, 8000 Aarhus C, Denmark

##  1\. Introduction

A flow on a $C^{*}$-algebra $A$ is a continuous one-parameter group
$\alpha=\left(\alpha_{t}\right)_{t\in\mathbb{R}}$ of automorphisms of $A$. In
quantum statistical mechanics such a flow is sometimes interpreted as the time
evolution of the observables of a physical system and the equilibrium states
of the system are then given by states $\phi$ of $A$ that satisfy the trace-
like condition

$\phi(ab)=\phi(b\alpha_{i\beta}(a))$

for all $a,b\in A$ with $a$ analytic for $\alpha$, [BR]. The number $\beta$
for which this holds is interpreted as the inverse temperature of the system
in the state given by $\phi$, and $\phi$ is said to be a $\beta$-KMS state for
$\alpha$. The $C^{*}$-algebra $A$ of observables in such a model is often a
UHF algebra and it is therefore of interest to determine the KMS states and
the possible inverse temperatures which can occur for flows on a UHF algebra.
It follows from work by Powers and Sakai in the 70’s, [PS], that when the flow
is approximately inner the set of possible inverse temperatures is the whole
real line $\mathbb{R}$, and Powers and Sakai conjectured that all flows on a
UHF algebra are approximately inner. That this is not the case was proved by
Matui and Sato in [MS], following work by Kishimoto on the AF algebra case,
[Ki2]. It is therefore an open question which sets of real numbers can occur
as the set of inverse temperatures for a flow on a UHF algebra. The set is
closed for any flow and since a UHF algebra has a unique trace state which is
automatically a $0$-KMS state, it must contain zero. These are the only known
restrictions. It is the purpose with this paper to show that it is also the
only restrictions, at least when one only considers non-negative $\beta$, and
not only for flows on UHF algebras, but in fact on all simple infinite
dimensional unital AF algebras.

In the examples of Kishimoto, and also in the examples of Matui and Sato, the
set of possible inverse temperatures is as small as it can be, consisting only
of $0$, cf. Remark 3.2 below. Here we will show that a variant of the method
developed in [Th5] can be used to construct a flow on any infinite dimensional
simple unital AF algebra such that the set of inverse temperatures is any
given lower bounded closed set of real numbers containing zero. For this we
use a method which is a natural extension of ideas and methods from work by
Bratteli, Elliott, Herman and Kishimoto in [BEH], [BEK],[Ki2], and by Matui
and Sato in [MS]. As in these papers we depend on results from the
classification of simple $C^{*}$-algebras. Besides the classification of AF
algebras, which was fundamental already in [BEH], we apply the recent work by
Castillejos, Evington, Tikuisis, White and Winter in [CETWW], where the
classification results we use in turn are based on work by Elliott, Gong, Lin
and Niu, [GLN], [EGLN], among others. However, when the algebra in question is
UHF there is an alternative route to the desired result which we describe
Remark 4.12.

The paper contains some material which appeared in [Th5]; this is because I
now consider [Th5] a forerunner to the present paper and it will not be
submitted for publication. The results of the present paper answer Question
2.6 in [Th5] positively, but Question 6.2 in [Th4] only partially. What
remains is to determine if it is possible to remove the condition that the set
is bounded below. This restriction is annoying and probably not necessary.
However, in applications and also to many mathematicians, the (inverse)
temperatures are non-negative and then the restriction to the lower bounded
case is of no significance.

In relation to this, one should observe the recent work by Christensen and
Vaes, [CV], in which they connect the theory of KMS states on $C^{*}$-algebras
to the theory of group actions on spaces. They also provide two examples of
simple unital $C^{*}$-algebras on which there are flows with arbitrary KMS
spectra. While it is certainly premature to suggest that this applies to all
infinite dimensional unital simple $C^{*}$-algebras, it does seem appropriate
to point out that we don’t really know.

_Acknowledgement_ I am grateful to Y. Sato for comments on the first version
of this paper which helped me navigate in the litterature on the
classification of simple $C^{*}$-algebras, and I thank the referee of [Th5]
for his remarks. The work was supported by the DFF-Research Project 2
‘Automorphisms and Invariants of Operator Algebras’, no. 7014-00145B.

##  2\. Statement of results

######  Theorem 2.1.

Let $A$ be a unital infinite dimensional simple AF $C^{*}$-algebra. For each
non-empty closed face $F$ in the tracial state space $T(A)$ of $A$ and for
each closed and lower bounded set $K\subseteq\mathbb{R}$ of real numbers
containing $0$ there is a $2\pi$-periodic flow $\alpha$ on $A$ such that

  * •

there is a $\beta$-KMS state for $\alpha$ if and only if $\beta\in K$,

  * •

for $\beta\in K\backslash\\{0\\}$ the simplex of $\beta$-KMS states for
$\alpha$ is affinely homeomorphic to $F$,

  * •

the simplex of $0$-KMS states for $\alpha$ is affinely homeomorphic to $T(A)$,
and

  * •

the fixed point algebra $A^{\alpha}$ of $\alpha$ is an $AF$ algebra.

When $K$ is also upper bounded, and hence compact, $\alpha$ can be chosen such
that the fixed point algebra $A^{\alpha}$ of $\alpha$ is a simple $AF$
algebra.

When $K$ is not compact it is not possible to arrange that $A^{\alpha}$ is
simple because when the set of inverse temperatures contains arbitrarily large
numbers the flow must have ground states, and this is not possible for
periodic flows whose fixed point algebra is simple, cf. Remark 3.5 in [BEH].

By taking $F$ to consist of a single extremal point in $T(A)$ we get the
following

######  Corollary 2.2.

Let $A$ be a unital infinite dimensional simple AF $C^{*}$-algebra. For each
closed and lower bounded set $K\subseteq\mathbb{R}$ of real numbers containing
$0$ there is a $2\pi$-periodic flow $\alpha$ on $A$ such that

  * •

there is a $\beta$-KMS state for $\alpha$ if and only if $\beta\in K$,

  * •

for $\beta\in K\backslash\\{0\\}$ the $\beta$-KMS state is unique,

  * •

the simplex of $0$-KMS states for $\alpha$ is a affinely homeomorphic to
$T(A)$, and

  * •

the fixed point algebra $A^{\alpha}$ of $\alpha$ is an $AF$ algebra.

When $K$ is also upper bounded, and hence compact, $\alpha$ can be chosen such
that the fixed point algebra $A^{\alpha}$ of $\alpha$ is a simple $AF$
algebra.

In the last section of the paper we combine Corollary 2.2 with methods and
results from [Th3] and [Th4] to obtain the following additional corollaries.
Recall that two compact Choquet simplexes are strongly affinely isomorphic
when there is an affine bijection between them which restricts to a
homeomorphism between the sets of extremal points. For Bauer simplexes this is
the same as affine homeomorphism, but in general it is a weaker notion. See
[Th4].

######  Corollary 2.3.

Let $U$ be a UHF algebra and let $K$ be a closed and lower bounded set of real
numbers containing $0$. Let $\mathbb{I}$ be a finite or countably infinite
collection of intervals in $\mathbb{R}$ such that $I=\mathbb{R}$ for at least
one $I\in\mathbb{I}$. For each $I\in\mathbb{I}$ choose a compact metrizable
Choquet simplex $S_{I}$ and for $\beta\in K$ set
$\mathbb{I}_{\beta}=\left\\{I\in\mathbb{I}:\ \beta\in I\right\\}$. There is a
$2\pi$-periodic flow $\alpha$ on $U$ such that

  * •

there is $\beta$-KMS state for $\alpha$ if and only if $\beta\in K$,

  * •

for each $I\in\mathbb{I}$ and $\beta\in(I\cap K)\backslash\\{0\\}$ there is a
closed face $F_{I}$ in $S^{\alpha}_{\beta}$ strongly affinely isomorphic to
$S_{I}$, and

  * •

for each $\beta\in K\backslash\\{0\\}$ and each $\omega\in S^{\alpha}_{\beta}$
there is a unique norm-convergent decomposition

$\omega=\sum_{I\in\mathbb{I}_{\beta}}\omega_{I}\ ,$

where $\omega_{I}\in\left\\{t\mu:\ t\in]0,1],\ \mu\in F_{I}\right\\}$.

######  Corollary 2.4.

Let $U$ be a UHF algebra and let $K$ be a closed and lower bounded set of real
numbers containing $0$. There is a $2\pi$-periodic flow $\alpha$ on $U$ such
that $S^{\alpha}_{\beta}=\emptyset$ if and only if $\beta\notin K$, and for
$\beta,\beta^{\prime}\in K$ the simplexes $S^{\alpha}_{\beta}$ and
$S^{\alpha}_{\beta^{\prime}}$ are not strongly affinely isomorphic unless
$\beta=\beta^{\prime}$.

##  3\. Preparations

In this paper all $C^{*}$-algebras are assumed to be separable and all traces
and weights on a $C^{*}$-algebra are required to be non-zero, densely defined
and lower semi-continuous. Concerning weights and in particular KMS weights we
shall use notation and results from Sections 1.1 and 1.3 in [KV]. Let $A$ be a
$C^{*}$-algebra and $\theta$ a flow on $A$. Let $\beta\in\mathbb{R}$. A
$\beta$-KMS weight for $\theta$ is a weight $\omega$ on $A$ such that
$\omega\circ\theta_{t}=\omega$ for all $t$, and

$\omega(a^{*}a)\ =\
\omega\left(\theta_{-\frac{i\beta}{2}}(a)\theta_{-\frac{i\beta}{2}}(a)^{*}\right)\
\ \ \forall a\in D(\theta_{-\frac{i\beta}{2}})\ .$ (3.1)

In particular, a $0$-KMS weight for $\theta$ is a $\theta$-invariant trace. It
was shown by Kustermans in Theorem 6.36 of [Ku] that this definition agrees
with the one introduced by Combes in [C]. It is because of the formulation
given by (3.1), which was not available when [BEH] was written, that we are
able to work with KMS weights throughout the present work. A bounded
$\beta$-KMS weight is called a $\beta$-KMS functional and a $\beta$-KMS state
when it is of norm $1$.

The first lemma can be considered as an updated version of a part of the
discussion in Remark 3.3 of [BEH].

######  Lemma 3.1.

Let $B$ be a $C^{*}$-algebra and $\gamma\in\operatorname{Aut}(B)$ an
automorphism of $B$. Let $\widehat{\gamma}$ be the dual action on
$B\rtimes_{\gamma}\mathbb{Z}$ considered as a $2\pi$-periodic flow. For
$\beta\in\mathbb{R}$ the restriction map $\omega\mapsto\omega|_{B}$ is a
bijection from the $\beta$-KMS weights for $\widehat{\gamma}$ onto the traces
$\tau$ on $B$ with the property that $\tau\circ\gamma=e^{-\beta}\tau$. The
inverse is the map $\tau\mapsto\tau\circ P$, where
$P:B\rtimes_{\gamma}\mathbb{Z}\to B$ is the canonical conditional expectation.

###### Proof.

Let $p_{1}\leq p_{2}\leq\cdots$ be an approximate unit in $B$ from the
Pedersen ideal $K(B)$ of $B$ with the additional property that $p_{k}^{2}\leq
p^{2}_{k+1}$ for all $k$, cf. [Pe]. Let $u$ be the canonical unitary
multiplier of $B\rtimes_{\gamma}\mathbb{Z}$ such that $ubu^{*}=\gamma(b)$ for
$b\in B$, and let $\tau$ be a trace on $B$ with the property that
$\tau\circ\gamma=e^{-\beta}\tau$. For $a,b\in B,\ n,m\in\mathbb{Z}$ we have
that

$P(p_{k}au^{n}p_{k}^{2}bu^{m}p_{k})=P(p_{k}bu^{m}p_{k}^{2}\widehat{\gamma}_{i\beta}(au^{n})p_{k})=0$

unless $m=-n$. Set $a^{\prime}=p_{k}a\gamma^{n}(p_{k}),\
b^{\prime}=p_{k}b\gamma^{-n}(p_{k})$. Then $a^{\prime},b^{\prime}\in K(B)$,
and by using Proposition 5.5.2 in [Pe] we find that

$\displaystyle\tau\circ
P(p_{k}bu^{-n}p_{k}\widehat{\gamma}_{i\beta}(p_{k}au^{n}p_{k}))=e^{-n\beta}\tau(p_{k}bu^{-n}p_{k}^{2}au^{n}p_{k})$
$\displaystyle=e^{-n\beta}\tau(b^{\prime}u^{-n}a^{\prime}u^{n})=\tau(\gamma^{n}(b^{\prime})a^{\prime})$
$\displaystyle=\tau(a^{\prime}\gamma^{n}(b^{\prime}))=\tau\circ
P(p_{k}au^{n}p_{k}^{2}bu^{-n}p_{k})\ ,$

showing that $\tau\circ P$ is a $\beta$-KMS functional on
$p_{k}(B\rtimes_{\gamma}\mathbb{Z})p_{k}$ for the restriction of
$\widehat{\gamma}$. We shall use repeatedly that when $b\geq 0$ in $B$ we have
that

$\lim_{k\to\infty}\tau(p_{k}bp_{k})=\tau(b)\ ,$ (3.2)

which follows from the lower semi-continuity of $\tau$ since
$\tau(p_{k}bp_{k})=\tau(\sqrt{b}p_{k}^{2}\sqrt{b})$ and
$\sqrt{b}p_{k}^{2}\sqrt{b}$ increases to $b$ as $k\to\infty$. Let $x\in
D(\widehat{\gamma}_{-\frac{i\beta}{2}})$. Then

$\displaystyle\tau\circ
P\left(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*}\right)=\lim_{k\to\infty}\tau\circ
P\left(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)p_{k}^{2}\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*}\right)$
$\displaystyle=\lim_{k\to\infty}\lim_{l\to\infty}\tau\left(p_{l}P\left(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)p_{k}^{2}\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*}\right)p_{l}\right)\
.$

We have shown above that $\tau\circ P$ is a $\beta$-KMS functional on
$p_{l}(B\rtimes_{\gamma}\mathbb{Z})p_{l}$ and when $l\geq k$ this gives

$\displaystyle\tau\left(p_{l}P\left(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)p_{k}^{2}\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*}\right)p_{l}\right)=\tau\circ
P\left(\widehat{\gamma}_{-\frac{i\beta}{2}}(p_{l}xp_{k})\widehat{\gamma}_{-\frac{i\beta}{2}}(p_{l}xp_{k})^{*}\right)$
$\displaystyle=\ \tau\circ P((p_{l}xp_{k})^{*}p_{l}xp_{k})=\tau\circ
P(p_{k}x^{*}p_{l}^{2}xp_{k})\leq\tau\left(p_{k}P(x^{*}x)p_{k}\right)\ .$

By using (3.2) we conclude that

$\tau\circ
P\left(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*}\right)\
\leq\ \tau\circ P(x^{*}x)\ .$

Similarly,

$\displaystyle\tau\circ P(x^{*}x)=\lim_{k\to\infty}\tau\circ
P(x^{*}p_{k}^{2}x)=\lim_{k\to\infty}\lim_{l\to\infty}\tau\circ
P(p_{l}x^{*}p_{k}^{2}xp_{l})\ .$

When $l\geq k$,

$\displaystyle\tau\circ P(p_{l}x^{*}p_{k}^{2}xp_{l})=\tau\circ
P(\widehat{\gamma}_{-\frac{i\beta}{2}}(p_{k}xp_{l})\widehat{\gamma}_{-\frac{i\beta}{2}}(p_{k}xp_{l})^{*})$
$\displaystyle=\tau\circ
P(p_{k}\widehat{\gamma}_{-\frac{i\beta}{2}}(x)p_{l}^{2}\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*}p_{k}\
\leq\tau\left(p_{k}P(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*})p_{k}\right),$

and we find therefore that

$\displaystyle\tau\circ
P(x^{*}x)\leq\lim_{k\to\infty}\tau\left(p_{k}P(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*})p_{k}\right)=\tau\circ
P(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*})\
.$

We conclude therefore first that $\tau\circ P(x^{*}x)=\tau\circ
P(\widehat{\gamma}_{-\frac{i\beta}{2}}(x)\widehat{\gamma}_{-\frac{i\beta}{2}}(x)^{*})$,
and then that $\tau\circ P$ is a $\beta$-KMS weight.

Let $\omega$ be a $\beta$-KMS weight for $\widehat{\gamma}$. Then
$\omega(b^{*}b)=\omega(bb^{*})$ for all $b\in B$ because
$\widehat{\gamma}_{-i\frac{\beta}{2}}(b)=b$. Using Riemann sum approximations
to the integral

$P(a)=(2\pi)^{-1}\int_{0}^{2\pi}\widehat{\gamma}_{t}(a)\ \mathrm{d}t\ ,$

it follows from the $\widehat{\gamma}$-invariance and lower semi-continuity of
$\omega$ that

$\omega\circ P(a)\leq\omega(a)$ (3.3)

when $a\geq 0$ in $B\rtimes_{\gamma}\mathbb{Z}$. In particular, $\omega|_{B}$
is densely defined since $\omega$ is, and we conclude that $\tau=\omega|_{B}$
is a trace on $B$. Let $a\geq 0$. Then

$\omega\circ\gamma(a)=\omega(uau^{*})=e^{-\beta}\omega\left(\widehat{\gamma}_{-\frac{i\beta}{2}}(u\sqrt{a})\widehat{\gamma}_{-\frac{i\beta}{2}}(u\sqrt{a})^{*}\right)=e^{-\beta}\omega(a)\
.$

It follows that $\tau\circ\gamma=e^{-\beta}\tau$. It remains now only to show
that $\omega=\omega\circ P$. Note that because
$\omega\circ\widehat{\gamma}_{t}=\omega$ we find for all $b\in B,\
n\in\mathbb{Z}$, that

$\omega(p_{k}bu^{n}p_{k})=\begin{cases}0,&\ n\neq 0\\\ \omega(p_{k}bp_{k}),&\
n=0\end{cases}\ =\ \omega(p_{k}P(bu^{n})p_{k})\ ,$

implying that $\omega(p_{k}\ \cdot\ p_{k})=\omega(p_{k}P(\ \cdot\ )p_{k})$.
Let $a\in B\rtimes_{\gamma}\mathbb{Z}$, $a\geq 0$. Since $\omega|_{B}$ is a
trace we can use (3.2) to conclude that

$\lim_{k\to\infty}\omega(p_{k}ap_{k})=\lim_{k\to\infty}\omega(p_{k}P(a)p_{k})=\
\omega(P(a))\ .$

Since $\lim_{k\to\infty}p_{k}ap_{k}=a$ the lower semi-continuity of $\omega$
implies now that $\omega(a)\leq\omega(P(a))$. Combined with (3.3) this yields
the desired conclusion that $\omega\circ P=\omega$. ∎

######  Remark 3.2.

It follows from Lemma 3.1 that when $B$ has a unique trace the dual action
will have no $\beta$-KMS weights for $\beta\neq 0$, and if in addition
$B\rtimes_{\gamma}\mathbb{Z}$ is simple the restriction of the dual action to
any corner $e(B\rtimes_{\gamma}\mathbb{Z})e$ given by a
$\widehat{\gamma}$-invariant non-zero projection $e$ will have no $\beta$-KMS
states for $\beta\neq 0$ by Remark 3.3 in [LN] or Theorem 2.4 in [Th2]. This
observation applies to the flows that were shown not to be approximately inner
in [Ki2] and [MS].

Taking $\beta=0$ in Lemma 3.1 we get a special case which we shall need. It is
probably known.

######  Corollary 3.3.

Let $B$ be a $C^{*}$-algebra and $\gamma\in\operatorname{Aut}(B)$ an
automorphism of $B$. The restriction map $\omega\mapsto\omega|_{B}$ is a
bijection from the $\widehat{\gamma}$-invariant traces $\omega$ on
$B\rtimes_{\gamma}\mathbb{Z}$ onto the $\gamma$-invariant traces on $B$. The
inverse is the map $\tau\mapsto\tau\circ P$, where
$P:B\rtimes_{\gamma}\mathbb{Z}\to B$ is the canonical conditional expectation.

We remark that in general $B\rtimes_{\gamma}\mathbb{Z}$ can have traces that
are not invariant under the dual action $\widehat{\gamma}$ and hence are not
determined by their restriction to $B$; also in cases where $B$ is UHF and
$B\rtimes_{\gamma}\mathbb{Z}$ is simple. For the present purposes it is
crucial that we can circumvent this issue thanks to the following lemma which
is suggested by the work of Matui and Sato in [MS]. In fact, most of it
appears implicitly in [MS].

Recall that a $C^{*}$-algebra $A$ is stable when $A\otimes\mathbb{K}\simeq A$,
where $\mathbb{K}$ is the $C^{*}$-algebra of compact operators on an infinite
dimensional separable Hilbert space. Recall also that a partially ordered
group $(G,G^{+})$ has large denominators when the following holds: For any
$a\in G^{+}$ and any $n\in\mathbb{N}$ there is an element $b\in G$ and an
$m\in\mathbb{N}$ such that $nb\leq a\leq mb$, cf. [Ni].

######  Lemma 3.4.

Let $B$ be a stable AF-algebra and $\gamma\in\operatorname{Aut}(B)$. Assume
that $K_{0}(B)$ has large denominators. There is an automorphism
$\gamma^{\prime}\in\operatorname{Aut}(B)$ such that

  1. a)

$\gamma^{\prime}_{*}=\gamma_{*}$ on $K_{0}(B)$,

  2. b)

the restriction map $\mu\ \mapsto\ \mu|_{B}$ is a bijection from traces $\mu$
on $B\rtimes_{\gamma^{\prime}}\mathbb{Z}$ onto the $\gamma^{\prime}$-invariant
traces on $B$,

  3. c)

$B\rtimes_{\gamma^{\prime}}\mathbb{Z}$ is $\mathcal{Z}$-stable; i.e
$(B\rtimes_{\gamma^{\prime}}\mathbb{Z})\otimes\mathcal{Z}\simeq
B\rtimes_{\gamma^{\prime}}\mathbb{Z}$ where $\mathcal{Z}$ denotes the Jiang-Su
algebra, [JS], and

  4. d)

$B\rtimes_{\gamma^{\prime}}\mathbb{Z}$ is stable.

###### Proof.

The first step is to show that there is an isomorphism $\phi:B\to
B\otimes\mathcal{Z}$ such that $\phi_{*}={(\operatorname{id}_{B}\otimes
1_{\mathcal{Z}})}_{*}$. Since $\mathcal{Z}$ is strongly self-absorbing in the
sense of Toms and Winter, [TW], it suffices for this to show that $B$ is
$\mathcal{Z}$-stable. By Corollary 3.4 in [TW] it is enough to show that $eBe$
is $\mathcal{Z}$-stable for any non-zero projection $e\in B$, and hence by
Theorem 2.3 in [TW] it is enough to show that $eBe$ is approximately
divisible. Now $K_{0}(eBe)$ has large denominators since we assume that
$K_{0}(B)$ has, and it follows therefore from Lemma 4.4 in [Th1] that for any
Bratteli diagram for $eBe$ the multiplicity matrices describing the embedding
of a given level into the following levels will have the property that the
minimum of the non-zero entries increases to infinity. This implies that $eBe$
is approximately divisible, cf. [BKR]. We deduce in this way the existence of
$\phi$.

For the second step we use [Sa] to obtain an automorphism $\theta$ of
$\mathcal{Z}$ with the weak Rohlin property; that is, with the property that
for each $k\in\mathbb{N}$ there is a sequence $\\{f_{n}\\}$ in $\mathcal{Z}$
such that

  * •

$0\leq f_{n}\leq 1_{\mathcal{Z}}$ for all $n$,

  * •

$\lim_{n\to\infty}f_{n}a-af_{n}=0$ for all $a\in\mathcal{Z}$,

  * •

$\lim_{n\to\infty}\theta^{j}(f_{n})f_{n}=0$ for $j=1,2,3,\cdots,k$, and

  * •

$\lim_{n\to\infty}\tau\left(1_{\mathcal{Z}}-\sum_{j=0}^{k}\theta^{j}(f_{n})\right)=0$
,

where $\tau$ is the trace state of $\mathcal{Z}$. From the first step, applied
twice, it follows that there is an isomorphism $\phi:B\to
B\otimes\mathcal{Z}\otimes\mathcal{Z}$ such that
$\phi_{*}=(\operatorname{id}_{B}\otimes 1_{\mathcal{Z}}\otimes
1_{\mathcal{Z}})_{*}$. Since $B$ is stable there is also a $*$-isomorphism
$\psi_{0}:B\to\mathbb{K}\otimes B$ such that
${\psi_{0}}_{*}=(e\otimes\operatorname{id}_{B})_{*}$, where $e$ is a minimal
non-zero projection in $\mathbb{K}$. Set

$\psi=\left(\psi_{0}\otimes\operatorname{id}_{\mathcal{Z}}\otimes\operatorname{id}_{\mathcal{Z}}\right)\circ\phi:B\to\mathbb{K}\otimes
B\otimes\mathcal{Z}\otimes\mathcal{Z}\ $

and

$\gamma^{\prime}=\psi^{-1}\circ\left(\operatorname{id}_{\mathbb{K}}\otimes\gamma\otimes\theta\otimes\operatorname{id}_{\mathcal{Z}}\right)\circ\psi\
.$

It follows from the Künneth theorem that
$\psi_{*}=(e\otimes\operatorname{id}_{B}\otimes 1_{\mathcal{Z}}\otimes
1_{\mathcal{Z}})_{*}$ and hence
$\left(\operatorname{id}_{\mathbb{K}}\otimes\gamma\otimes\theta\otimes\operatorname{id}_{\mathcal{Z}}\right)_{*}\circ\psi_{*}=(e\otimes\gamma\otimes
1_{\mathcal{Z}}\otimes 1_{\mathcal{Z}})_{*}$. Thus
$\gamma^{\prime}_{*}=\gamma$. To complete the proof it suffices to verify that
b), c) and d) hold when $B$ is replaced by $\mathbb{K}\otimes
B\otimes\mathcal{Z}\otimes\mathcal{Z}$ and $\gamma^{\prime}$ by
$\operatorname{id}_{\mathbb{K}}\otimes\gamma\otimes\theta\otimes\operatorname{id}_{\mathcal{Z}}$.
The properties c) and d) follow immediately because
$\mathcal{Z}\otimes\mathcal{Z}\simeq\mathcal{Z}$,
$\mathbb{K}\otimes\mathbb{K}\simeq\mathbb{K}$ and

$(\mathbb{K}\otimes
B\otimes\mathcal{Z}\otimes\mathcal{Z})\rtimes_{\operatorname{id}_{\mathbb{K}}\otimes\gamma\otimes\theta\otimes\operatorname{id}_{\mathcal{Z}}}\mathbb{Z}\
\simeq\
\mathbb{K}\otimes\left((B\otimes\mathcal{Z})\rtimes_{\gamma\otimes\theta}\mathbb{Z}\right)\otimes\mathcal{Z}\
.$

In the final step we verify b). For this, set $B^{\prime}=\mathbb{K}\otimes
B\otimes\mathcal{Z}$ and
$\gamma^{\prime\prime}=\operatorname{id}_{\mathbb{K}}\otimes\gamma\otimes\operatorname{id}_{\mathcal{Z}}$.
Then

$(\mathbb{K}\otimes
B\otimes\mathcal{Z}\otimes\mathcal{Z})\rtimes_{\operatorname{id}_{\mathbb{K}}\otimes\gamma\otimes\theta\otimes\operatorname{id}_{\mathcal{Z}}}\mathbb{Z}\
\simeq\
(B^{\prime}\otimes\mathcal{Z})\rtimes_{\gamma^{\prime\prime}\otimes\theta}\mathbb{Z}\
,$

and it suffices to verify that b) holds when $B$ is replaced by
$B^{\prime}\otimes\mathcal{Z}$ and $\gamma^{\prime}$ by
$\gamma^{\prime\prime}\otimes\theta$. Let
$P:(B^{\prime}\otimes\mathcal{Z})\rtimes_{\gamma^{\prime\prime}\otimes\theta}\mathbb{Z}\to
B^{\prime}\otimes\mathcal{Z}$ be the canonical conditional expectation. In
view of Corollary 3.3 what remains is to consider a trace $\mu$ on
$(B^{\prime}\otimes\mathcal{Z})\rtimes_{\gamma^{\prime\prime}\otimes\theta}\mathbb{Z}$
and show that $\mu=\mu\circ P$. For this, let $\\{q_{n}\\}$ be an approximate
unit in $B^{\prime}$ consisting of projections. Since $q_{n}\otimes
1_{\mathcal{Z}}$ is a projection and therefore contained in the Pedersen
ideal, $\mu(q_{n}\otimes 1_{\mathcal{Z}})<\infty$, cf. [Pe]. When $a\geq 0$ in
$(B^{\prime}\otimes\mathcal{Z})\rtimes_{\gamma^{\prime\prime}\otimes\theta}\mathbb{Z}$,

$\mu(a)=\lim_{n}\mu\left(\sqrt{a}(q_{n}\otimes
1_{\mathcal{Z}})\sqrt{a}\right)=\lim_{n}\mu\left((q_{n}\otimes
1_{\mathcal{Z}})a(q_{n}\otimes 1_{\mathcal{Z}})\right)\ ,$

and similarly $\mu(P(a))=\lim_{n}\mu\left((q_{n}\otimes
1_{\mathcal{Z}})P(a)(q_{n}\otimes 1_{\mathcal{Z}})\right)$. It suffices
therefore to show that

$\mu\left((q_{n}\otimes 1_{\mathcal{Z}})a(q_{n}\otimes
1_{\mathcal{Z}})\right)=\mu\left((q_{n}\otimes
1_{\mathcal{Z}})P(a)(q_{n}\otimes 1_{\mathcal{Z}})\right)$

for all $n$. For this note that $x\mapsto\mu\left((q_{n}\otimes
1_{\mathcal{Z}})x(q_{n}\otimes 1_{\mathcal{Z}})\right)$ extends to a bounded
positive linear functional of norm $\mu\left(q_{n}\otimes
1_{\mathcal{Z}}\right)$ on
$(B^{\prime}\otimes\mathcal{Z})\rtimes_{\gamma^{\prime\prime}\otimes\theta}\mathbb{Z}$.
It suffices therefore to consider positive elements $b\in B^{\prime}$,
$z\in\mathcal{Z}$ and $k\in\mathbb{Z}\backslash\\{0\\}$, and show that

$\mu((q_{n}\otimes 1_{\mathcal{Z}})(b\otimes z)u^{k}(q_{n}\otimes
1_{\mathcal{Z}}))=0$ (3.4)

when $u$ is the canonical unitary in the multiplier algebra of
$(B^{\prime}\otimes\mathcal{Z})\rtimes_{\gamma^{\prime\prime}\otimes\theta}\mathbb{Z}$
such that $\operatorname{Ad}u=\gamma^{\prime\prime}\otimes\theta$ on
$B^{\prime}\otimes\mathcal{Z}$. Since the complex conjugate of
$\mu((q_{n}\otimes 1_{\mathcal{Z}})(b\otimes z)u^{-k}(q_{n}\otimes
1_{\mathcal{Z}}))$ is $\mu((q_{n}\otimes
1_{\mathcal{Z}})({\gamma^{\prime\prime}}^{k}(b)\otimes\theta^{k}(z))u^{k}(q_{n}\otimes
1_{\mathcal{Z}}))$ we may assume $k\geq 1$. Let $\epsilon>0$. Since
$a\mapsto\mu(q_{n}\otimes a)$ is a bounded trace on $\mathcal{Z}$ it must be a
scalar multiple of the unique trace state $\tau$ of $\mathcal{Z}$. It follows
therefore from the properties of $\theta$ that there are elements $0\leq
g_{j}\leq 1_{\mathcal{Z}},\ j=0,1,2,\cdots,k$, in $\mathcal{Z}$ such that

$\displaystyle\left\|g_{j}z-zg_{j}\right\|\leq\epsilon\ \text{for all}\ j\ ,$
(3.5) $\displaystyle\left\|g_{i}g_{j}\right\|\leq\epsilon\ \text{for all}\
i,j,\ i\neq j,$ (3.6)
$\displaystyle\left\|\theta^{k}(g_{j})g_{j}\right\|\leq\epsilon\ \text{for all
}\ j,\ \text{and}$ (3.7)
$\displaystyle\left|\mu(q_{n}\otimes(1_{\mathcal{Z}}-\sum_{j=0}^{k}g_{j}))\right|\
\leq\ \epsilon\mu(q_{n}\otimes 1_{\mathcal{Z}}).\ $ (3.8)

In the following, when $s$ and $t$ are complex numbers such that for all
$\delta>0$ the $\epsilon$ in (3.5)-(3.8) can be chosen so small that
$|s-t|\leq\delta$, we will write $s\sim t$. Set

$y=1_{\mathcal{Z}}-\sum_{j=0}^{k}g_{j}\ .$

It follows from (3.5) that

$\mu((q_{n}\otimes 1_{\mathcal{Z}})(b\otimes zy)u^{k}(q_{n}\otimes
1_{\mathcal{Z}}))\ \sim\ \mu((q_{n}\otimes 1_{\mathcal{Z}})(b\otimes
yz)u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))\ ,$

and from the Cauchy-Schwarz inequality that

$\begin{split}&\left|\mu((q_{n}\otimes 1_{\mathcal{Z}})(b\otimes
yz)u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))\right|\ =\ \left|\mu((q_{n}\otimes
y)(b\otimes z)u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))\right|\\\ &\leq\
\sqrt{\mu(q_{n}\otimes y^{2})}\|b\|\|z\|\sqrt{\mu(q_{n}\otimes
1_{\mathcal{Z}})}\ .\end{split}$ (3.9)

Note that it follows from (3.6) that
$\left\|(\sum_{j=0}^{k}g_{j})^{2}-\sum_{j=0}^{k}g_{j}^{2}\right\|\leq(k+1)^{2}\epsilon$
and hence

$(\sum_{j=0}^{k}g_{j})^{2}\leq\sum_{j=0}^{k}g_{j}^{2}+(k+1)^{2}\epsilon
1_{\mathcal{Z}}\ \leq\ \sum_{j=0}^{k}g_{j}+(k+1)^{2}\epsilon 1_{\mathcal{Z}}\
.$

It follows that $y^{2}\leq y+(k+1)^{2}\epsilon 1_{\mathcal{Z}}$ and then from
(3.8) that

$0\ \leq\ \mu(q_{n}\otimes y^{2})\ \leq\ ((k+1)^{2}+1)\mu(q_{n}\otimes
1_{\mathcal{Z}})\epsilon\ .$

Combined with (3.9) this shows that $\mu((q_{n}\otimes
1_{\mathcal{Z}})(b\otimes yz)u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))\sim 0$, so
to obtain (3.4) it suffices to show that $\mu((q_{n}\otimes
1_{\mathcal{Z}})(b\otimes zg_{j}))u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))\sim 0$
for each $j$. It follows from (3.5) that

$\mu((q_{n}\otimes 1_{\mathcal{Z}})(b\otimes zg_{j}))u^{k}(q_{n}\otimes
1_{\mathcal{Z}}))\ \sim\ \mu((q_{n}\otimes
1_{\mathcal{Z}})(b\otimes\sqrt{g_{j}}z\sqrt{g_{j}}))u^{k}(q_{n}\otimes
1_{\mathcal{Z}}))\ .$

Using Proposition 5.5.2 in [Pe] for the second equality we get that

$\displaystyle\mu((q_{n}\otimes
1_{\mathcal{Z}})(b\otimes\sqrt{g_{j}}z\sqrt{g_{j}})u^{k}(q_{n}\otimes
1_{\mathcal{Z}}))$ $\displaystyle=\mu((q_{n}\otimes\sqrt{g_{j}})(b\otimes
z\sqrt{g_{j}})u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))$
$\displaystyle=\mu((q_{n}\otimes 1_{\mathcal{Z}})(b\otimes
z\sqrt{g_{j}})u^{k}(q_{n}\otimes\sqrt{g_{j}}))$
$\displaystyle=\mu((q_{n}\otimes
1_{\mathcal{Z}})(b{\gamma^{\prime\prime}}^{k}(q_{n})\otimes
z\sqrt{g_{j}}\theta^{k}(\sqrt{g_{j}}))u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))\ .$

It follows therefore from (3.7) that $\mu((q_{n}\otimes
1_{\mathcal{Z}})(b\otimes zg_{j}))u^{k}(q_{n}\otimes 1_{\mathcal{Z}}))\sim 0$
as desired. ∎

The next lemma is well-known.

######  Lemma 3.5.

Let $B$ be an AF algebra. There is a bijective correspondence between traces
$\tau$ on $B$ and the set of non-zero positive homomorphisms
$K_{0}(B)\to\mathbb{R}$. The bijection is given by the formula
$\tau_{*}[e]=\tau(e)$ when $e$ is a projection in $B$.

##  4\. Proof of Theorem 2.1 when $K$ is compact

The case where $K$ is compact is sufficiently different from the case where
this is not the case, that we divide the proof accordingly. However, the
general setup which we now describe will be the same in the two cases.

###  4.1. Setting the stage

Let $A$ be an infinite dimensional simple unital AF algebra. The tracial state
space $T(A)$ of $A$ is a compact metrizable Choquet simplex which we denote by
$\Delta$. We shall handle $\Delta$ as an abstract simplex and when an element
$x\in\Delta$ is considered as a trace on $A$ we denote it by
$\operatorname{tr}_{x}$. The $K_{0}$-group $K_{0}(A)$ is a simple non-cyclic
dimension group and we denote by $K_{0}(A)^{+}$ the semi-group of positive
elements of $K_{0}(A)$. The unit $1$ in $A$ represents an element $[1]\in
K_{0}(A)$ which is an order unit in $K_{0}(A)$. To simplify notation we set

$(K_{0}(A),K_{0}(A)^{+},[1])\ =\ (H,H^{+},u)\ .$

Each element $x\in\Delta$ defines in a canonical way a homomorphism
${\operatorname{tr}_{x}}_{*}:H\to\mathbb{R}$, and since $A$ is AF the state
space

$\left\\{\phi\in\operatorname{Hom}(H,\mathbb{R}):\
\phi(H^{+})\subseteq[0,\infty),\ \phi(u)=1\right\\}$

of $(H,H^{+},u)$ is affinely homeomorphic to $\Delta$ via the map $\Delta\ni
x\mapsto{\operatorname{tr}_{x}}_{*}$. Let ${\operatorname{Aff}}(\Delta)$
denote the space of real-valued continuous affine functions on $\Delta$. There
is then a homomorphism

$\theta:H\to{\operatorname{Aff}}(\Delta)$

defined such that $\theta(h)(x)={\operatorname{tr}_{x}}_{*}(h)$. By Theorem
4.11 in [GH],

  * •

$\theta(H)$ is norm-dense in ${\operatorname{Aff}}(\Delta)$, and

  * •

$H^{+}\ =\ \left\\{h\in H:\ \theta(h)(x)>0\ \ \forall
x\in\Delta\right\\}\cup\\{0\\}$ .

Set

$G\ =\ \oplus_{\mathbb{Z}}H\ ,$

and define an automorphism $\rho$ of $G$ such that

$\rho\left((g_{n})_{n\in\mathbb{Z}}\right)\
=\left(g_{n+1}\right)_{n\in\mathbb{Z}}\ .$ (4.1)

The main difference between the two cases, compact and not compact, is in the
definition of the ordering in $G$, and although many arguments are the same in
the two cases, and will only be given once, we make a clear distinction
between the two cases in order to increase the readability.

###  4.2. The first construction

We assume now that the set $K$ from Theorem 2.1 is compact and set
$L=\left\\{e^{\beta}:\ \beta\in K\right\\}$. Let $A(\Delta\times L)$ denote
the real vector space of functions $f\in C_{\mathbb{R}}(\Delta\times L)$ for
which the map $\Delta\ni x\mapsto f(x,t)$ is affine for all $t\in L$; clearly
a norm-closed subspace of $C_{\mathbb{R}}(\Delta\times L)$. Define
$\Sigma:G\to A(\Delta\times L)$ such that

$\Sigma\left(g\right)(x,t)\ =\ \sum_{m\in\mathbb{Z}}\theta(g_{m})(x)t^{m}\ .$

Let $F$ be a non-empty closed face in $\Delta$ and set

$G^{+}\ =\ \left\\{g\in G:\ \Sigma(g)(x,t)\ >\ 0\ \ \forall(x,t)\in(F\times
L)\cup(\Delta\times\\{1\\})\right\\}\cup\\{0\\}\ ;$

a semi-group in $G$ which turns $G$ into a partially ordered abelian group. We
aim to show that $G$ is a simple dimension group.

For $(k,l)\in\mathbb{Z}^{2}$ let $g_{k,l}\in C_{\mathbb{R}}(L)$ be the
function

$g_{k,l}(t)\ =\ t^{k}-t^{l}\ .$

######  Lemma 4.1.

$\operatorname{Span}\\{g_{k,l}:\ (k,l)\in\mathbb{Z}^{2}\\}$ is dense in
$\left\\{h\in C_{\mathbb{R}}(L):\ h(1)=0\right\\}$.

###### Proof.

Note that $tg_{k,l}(t)=g_{k+1,l+1}(t)$. It follows that
$\operatorname{Span}\\{g_{k,l}:\ (k,l)\in\mathbb{Z}^{2}\\}$ contains all
functions of the form $P(t)\left(t-t^{-1}\right)$ where $P$ is a polynomial.
It follows therefore from Weierstrass’ theorem that the closure of
$\operatorname{Span}\\{g_{k,l}:\ (k,l)\in\mathbb{Z}^{2}\\}$ contains all
functions in $C_{\mathbb{R}}(L)(t-t^{-1})$. Let $h\in C_{\mathbb{R}}(L)$
satisfy that $h(1)=0$, and let $\epsilon>0$. There is a function $h_{1}\in
C_{\mathbb{R}}(L)$ such that $h_{1}(t)=0$ for all $t$ in a neighborhood of $1$
and $\sup_{t\in L}|h(t)-h_{1}(t)|\leq\epsilon$. Since

$h_{1}(t)=\frac{h_{1}(t)}{t-t^{-1}}(t-t^{-1})$

it follows first that $h_{1}$ is in the closure of
$\operatorname{Span}\\{g_{k,l}:\ (k,l)\in\mathbb{Z}^{2}\\}$, and then that so
is $h$. ∎

When $h\in H$ and $k,l\in\mathbb{Z},\ k\neq l$, we denote in the following by
$[[h]]^{k,l}$ the element of $G$ such that

$\left([[h]]^{k,l}\right)_{i}\ =\ \begin{cases}h,&\ i=k\\\ -h,&\ i=l\\\ 0,&\
i\notin\\{k,l\\}\ ,\end{cases}$

and by $[[h]]\in G$ the element with $[[h]]_{0}=h$ and $[[h]]_{i}=0$ when
$i\neq 0$.

######  Lemma 4.2.

$\Sigma(G)$ is dense in $A(\Delta\times L)$.

###### Proof.

Note that $\Sigma\left([[h]]^{k,l}\right)(x,t)\ =\ \theta(h)(x)g_{k,l}(t)$ and
$\Sigma([[h]])(x,t)=\theta(h)(x)$. Hence $\Sigma(G)$ contains all functions of
the form

$\theta(h)(x)(t^{k}-t^{l})\ ,$

where $k,l\in\mathbb{Z}$, as well as all functions of the form $\theta(h)(x)$
when $h\in H$. Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ it
follows from Lemma 4.1 that the closure $\overline{\Sigma(G)}$ of $\Sigma(G)$
contains all functions of the form $(x,t)\mapsto a(x)f(t)$ where
$a\in{\operatorname{Aff}}(\Delta)$ and $f\in C_{\mathbb{R}}(L)$. A wellknown
partition of unity argument shows then that $A(\Delta\times
L)=\overline{\Sigma(G)}$. ∎

######  Lemma 4.3.

Let $a\in{\operatorname{Aff}}(F)$. For each $x_{0}\in\Delta\backslash F$ there
is a function $b\in{\operatorname{Aff}}(\Delta)$ such that $b|_{F}=a$ and
$b(x_{0})>a(y)$ for all $y\in F$.

###### Proof.

Choose $u,v\in\mathbb{R}$ such that $u\leq a(y)\leq v$ for all $y\in F$. By
the Hahn-Banach separation theorem there is a
$c\in{\operatorname{Aff}}(\Delta)$ such that $c(y)\leq 0$ for all $y\in F$ and
$c(x_{0})>0$. Multiplying by a positive number we can arrange that
$c(x_{0})>v-u$. Set $C=\sup_{x\in\Delta}c(x)$. Then $c(y)\leq a(y)-u\leq
v-u+C$ for all $y\in F$ and $c(x)\leq v-u+C$ for all $x\in\Delta$. It follows
from Edwards separation theorem, Theorem 3 in [Ed] or Corollary 7.7 in [AE],
that there is a $b^{\prime}\in{\operatorname{Aff}}(\Delta)$ such that
$b^{\prime}|_{F}=a-u$ and $c\leq b^{\prime}\leq v-u+C$ on $\Delta$. Set
$b=b^{\prime}+u$. Then $b|_{F}=a$ and $b(x_{0})\geq c(x_{0})+u>v$. ∎

######  Lemma 4.4.

$G$ is a simple dimension group.

###### Proof.

It is easy to see that $G$ is unperforated and that
$G^{+}\cap(-G^{+})=\\{0\\}$. Since $(F\times L)\cup(\Delta\times\\{1\\})$ is
compact it is also clear that every non-zero element of $G^{+}$ is an order
unit for $G$, so what remains is to show that $G$ has the Riesz interpolation
property. Let $c^{i},d^{i},\ i\in\\{1,2\\}$, be elements of $G$ such that
$c^{i}\leq d^{j}$ for all $i,j\in\\{1,2\\}$. If
$c^{i^{\prime}}=d^{j^{\prime}}$ for some $i^{\prime},j^{\prime}$, set
$z=c^{i^{\prime}}$. Then $c^{i}\leq z\leq d^{j}$ for all $i,j$. Assume instead
that $c^{i}<d^{j}$ for all $i,j$. Then $\Sigma(c^{i})(x,t)<\Sigma(d^{j})(x,t)$
for all $i,j$ and all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$. Define
$c,d\in C_{\mathbb{R}}(\Delta\times L)$ such that

$c(x,t)\ =\ \max\\{\Sigma(c^{1})(x,t),\Sigma(c^{2})(x,t)\\}$

and

$d(x,t)\ =\ \min\\{\Sigma(d^{1})(x,t),\Sigma(d^{2})(x,t)\\}\ .$

Let $\delta>0$. If $\delta$ is small enough

$c(x,t)+\delta\ <\ d(x,t)-\delta$

for all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$. Fix $t\in
L\backslash\\{1\\}$. As a closed face in the simplex $\Delta$ the set $F$ is
itself a Choquet simplex and hence the space ${\operatorname{Aff}}(F)$ has the
Riesz interpolation property for the usual order, as well as for the strict
order, cf. Lemma 3.1 in [EHS]. It follow therefore that there is a function
$a_{t}\in{\operatorname{Aff}}(F)$ such that

$c(y,t)+\delta<a_{t}(y)<d(y,t)-\delta$

for all $y\in F$. When $t=1$ it follows in the same way that there is a
function $a_{1}\in{\operatorname{Aff}}(\Delta)$ such that

$c(x,1)+\delta\ <\ a_{1}(x)\ <\ d(x,1)-\delta$

for all $x\in\Delta$. We can then construct a finite cover
$V_{i},i=1,2,\cdots,N$, of $L$ by open sets and elements $t_{i}\in V_{i}$ with
$t_{1}=1$ such that

$\left|c(x,t)-c(x,t_{i})\right|\leq\frac{\delta}{2}$

and

$\left|d(x,t)-d(x,t_{i})\right|\leq\frac{\delta}{2}$

for all $t\in V_{i}$, all $x\in\Delta$ and all $i$. We arrange, as we can,
that $1\notin V_{j},\ j\geq 2$. For $i\in\\{2,3,\cdots,N\\}$ we use Lemma 4.3
to get $b_{i}\in{\operatorname{Aff}}(\Delta)$ such that
$b_{i}|_{F}=a_{t_{i}}$, and we set $b_{1}=a_{1}$. Let
$\\{\varphi_{i}\\}_{i=1}^{N}$ be a partition of unity on $L$ which is
subordinate to $\\{V_{i}\\}_{i=1}^{N}$. Define $a\in A(\Delta\times L)$ such
that

$a(x,t)\ =\ \sum_{i=1}^{N}b_{i}(x)\varphi_{i}(t)\ .$

For $y\in F$,

$\displaystyle a(y,t)\ =\ \sum_{i=1}^{N}a_{t_{i}}(y)\varphi_{i}(t)\ \leq\
\sum_{i=1}^{N}(d(y,t_{i})-\delta)\varphi_{i}(t)\ $ $\displaystyle\leq\
\sum_{i=1}^{N}(d(y,t)-\frac{\delta}{2})\varphi_{i}(t)\ =\
d(y,t)-\frac{\delta}{2}\ ,$

and similarly $c(y,t)+\frac{\delta}{2}\ \leq\ a(y,t)$. For $x\in\Delta$ we
find that

$\displaystyle c(x,1)+\delta\ <\ a_{1}(x)\ =\ a(x,1)\ <\ d(x,1)-\delta\ .$

It follows that

$\Sigma(c^{i})(x,t)<a(x,t)<\Sigma(d^{j})(x,t)$

for all $i,j$ and all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$. Let
$\epsilon>0$. By Lemma 4.2 there is a $g\in G$ such that

$\sup_{(x,t)\in\Delta\times L}\left|\Sigma(g)(x,t)\ -\ a(x,t)\right|\ \leq\
\epsilon\ .$

If $\epsilon$ is small enough it follows that $c^{i}\leq g\leq d^{j}$ in $G$
for all $i,j\in\\{1,2\\}$. ∎

Note that $\rho(G^{+})=G^{+}$ where $\rho$ is the automorphism $\rho$ of $G$
defined in (4.1). It follows that $\rho$ is an automorphism of $(G,G^{+})$ and
then from Lemma 4.4 and [EHS] that there is an AF algebra $B$ whose
$K_{0}$-group and dimension range is isomorphic to $(G,G^{+})$. Furthermore,
it follows from [E1] that $B$ is simple and stable, and that there is an
automorphism $\gamma$ of $B$ such that $\gamma_{*}=\rho$ under the
identification $K_{0}(B)=G$. It was shown by Nistor, [Ni], that $(G,G^{+})$
has large denominators and we can therefore use Lemma 3.4 to choose $\gamma$
such that it has the following additional properties:

######  Additional properties 4.5.

  1. (1)

The restriction map $\mu\ \mapsto\ \mu|_{B}$ is a bijection from the traces
$\mu$ on $B\rtimes_{\gamma}\mathbb{Z}$ onto the $\gamma$-invariant traces on
$B$.

  2. (2)

$B\rtimes_{\gamma}\mathbb{Z}$ is stable.

  3. (3)

$B\rtimes_{\gamma}\mathbb{Z}$ is $\mathcal{Z}$-stable.

Set $C=B\rtimes_{\gamma}\mathbb{Z}$. No power of $\gamma$ is inner since
$\gamma_{*}^{k}\neq\operatorname{id}_{G}$ for $k\neq 0$ and it follows
therefore from [Ki1] that $C$ is simple. The Pimsner-Voiculescu exact sequence
shows that $K_{1}(C)=0$ since $\operatorname{id}_{G}-\gamma_{*}$ is injective
and that

$K_{0}(C)\ =\ \operatorname{coker}(\operatorname{id}_{G}-\gamma_{*})\ =\
G/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)\ .$

Under this identification the map $K_{0}(B)\to K_{0}(C)$ induced by the
inclusion $B\subseteq C$ is the quotient map $q:G\to
G/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)$. Hence
$G^{+}/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)\subseteq
K_{0}(C)^{+}$.

######  Lemma 4.6.

Let $d\in G$. Then $\sum_{m\in\mathbb{Z}}d_{m}=0$ if and only if
$d=(\operatorname{id}_{G}-\gamma_{*})\left(g\right)$ for some $g\in G$.

###### Proof.

It obvious that $\sum_{m\in\mathbb{Z}}d_{m}=0$ when
$d=(\operatorname{id}_{G}-\gamma_{*})\left(g\right)$ for some $g\in G$. For
the converse, choose $L\in\mathbb{N}$ so big that $x_{n}=0$ when $|n|\geq L$.
Set $y_{n}=0$ when $n>L$ and

$y_{L}=x_{L},\ y_{L-1}=x_{L-1}+y_{L},\ y_{L-2}=x_{L-2}+y_{L-1},\ \text{etc}.$

Then

$y_{-L}=x_{-L}+\sum_{i=1}^{2L}x_{-L+i}=\sum_{i=1}^{2L}x_{-L+i}=0\ ,$

and hence $y_{k}=0$ when $k<-L$. It follows that $(y_{n})_{n\in\mathbb{Z}}\in
G$ and
$(\operatorname{id}_{G}-\gamma_{*})\left((y_{n})_{n\in\mathbb{Z}}\right)=(x_{n})_{n\in\mathbb{Z}}$.
∎

It follows from Lemma 4.6 that we can define an injective homomorphism
$S:G/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)\to H$ such that

$S\left(q(g)\right)\ =\ \sum_{m\in\mathbb{Z}}g_{m}\ \ \ \ \forall g\in G\ .$

$S$ is surjective since $S(q([[h]]))=h$ when $h\in H$, and hence an
isomorphism with inverse $S^{-1}$ given by $S^{-1}(h)=q([[h]])$. Let $p\in B$
be a projection such that $[p]=[[u]]$ in $G$.

######  Lemma 4.7.

For $(y,t)\in F\times L$ and $x\in\Delta$ there are traces $\tau_{y,t}$ and
$\tau_{x}$ on $B$ such that $\tau_{y,t}(p)=\tau_{x}(p)=1$, and

${\tau_{y,t}}_{*}(g)\ =\ \sum_{m\in\mathbb{Z}}\theta(g_{m})(y)t^{m}$

and

${\tau_{x}}_{*}(g)=\ \sum_{m\in\mathbb{Z}}\theta(g_{m})(x)$

for all $g\in G$.

###### Proof.

This follows from Lemma 3.5. ∎

######  Lemma 4.8.

Let $\phi:G\to\mathbb{R}$ be a positive homomorphism such that
$\phi([[u]])=1$. Assume that $\phi\circ\gamma_{*}=s^{-1}\phi$ for some $s>0$.
Then $s\in L$ and when $s\neq 1$ it follows that $\phi={\tau_{y,s}}_{*}$ for
some $y\in F$, and when $s=1$ it follows that $\phi={\tau_{x}}_{*}$ for some
$x\in\Delta$.

###### Proof.

We claim that there is continuous linear map $\phi^{\prime}:A(\Delta\times
L)\to\mathbb{R}$ such that $\phi=\phi^{\prime}\circ\Sigma$. Since $\theta(H)$
is dense in ${\operatorname{Aff}}(\Delta)$ there is a $c\in H$ such that
$0<\theta(c)(x)<N^{-1}$ for all $x\in\Delta$. Then $0\leq N[[c]]\leq[[u]]$ in
$G$ and hence
$0\leq\phi([[c]])\leq\frac{1}{N}\phi\left([[u]]\right)=\frac{1}{N}$. Assume
$g\in G$ and that $\Sigma(g)=0$. Then $\pm g+[[c]]\in G^{+}$ and hence
$-\frac{1}{N}\leq\phi(g)\leq\frac{1}{N}$. Letting $N\to\infty$ we conclude
that $\phi(g)=0$, and it follows that there is a homomorphism
$\phi^{\prime}:\Sigma(G)\to\mathbb{R}$ such that
$\phi^{\prime}\circ\Sigma=\phi$. Let $h\in\Sigma(G)$; say $h=\Sigma(g)$, and
let $k,l\in\mathbb{N}$ be natural numbers such that $|h(x,t)|<\frac{k}{l}$ for
all $(x,t)\in\Delta\times L$. Then $k[[u]]\pm lg\in G^{+}$ and hence

$0\ \leq\ \phi(k[[u]]\pm lg)\ =\ k\pm l\phi^{\prime}(h)\ .$

It follows that $\left|\phi^{\prime}(h)\right|\leq\frac{k}{l}$, proving that
$\phi^{\prime}$ is Lipshitz continuous. Since $\Sigma(G)$ is dense in
$A(\Sigma\times L)$ by Lemma 4.2 it follows that $\phi^{\prime}$ extends by
continuity to a continuous linear map $\phi^{\prime}:A(\Delta\times
L)\to\mathbb{R}$, proving the claim. Let $T:A(\Delta\times L)\to
A(\Delta\times L)$ denote the operator

$T(\psi)(x,t)=t^{-1}\psi(x,t)\ .$

Since $\Sigma\circ\gamma_{*}(g)(x,t)\ =\ t^{-1}\Sigma(g)(x,t)$ for all $g\in
G$ and all $(x,t)\in\Delta\times L$, we find that

$s^{-1}\phi^{\prime}(\Sigma(g))\ =\ s^{-1}\phi(g)\ =\ \phi(\gamma_{*}(g))\ =\
\phi^{\prime}\circ\Sigma\circ\gamma_{*}(g))\ =\
\phi^{\prime}\left(T(\Sigma(g))\right)\ .$

It follows therefore from Lemma 4.2 that

$s^{-1}\phi^{\prime}=\phi^{\prime}\circ T\ .$ (4.2)

When $a\in{\operatorname{Aff}}(\Delta)$ and $f\in C_{\mathbb{R}}(L)$ we denote
by $a\otimes f$ the function $\Delta\times L\ni(x,t)\mapsto a(x)f(t)$. Assume
$a(x)\geq 0$ for all $x\in\Delta$ and that $f(t)\geq 0$ for all $t\in L$. It
follows then from Lemma 4.2 and the definition of $G^{+}$ that $a\otimes f$
can be approximated by elements from $\Sigma(G^{+})$, implying that
$\phi^{\prime}(a\otimes f)\geq 0$. There is therefore a bounded Borel measure
$\mu_{a}$ on $L$ such that

$\phi^{\prime}(a\otimes f)\ =\ \int_{L}f\ \mathrm{d}\mu_{a}\ \ \forall f\in
C_{\mathbb{R}}(L)\ .$

It follows from (4.2) that

$\int_{L}s^{-1}f(t)\ \mathrm{d}\mu_{a}(t)\ =\ \int_{L}t^{-1}f(t)\
\mathrm{d}\mu_{a}(t)$

for all $f\in C_{\mathbb{R}}(L)$, and hence that
$\mu_{a}(L\backslash\\{s\\})=0$. Since $\phi\neq 0$, not all $\mu_{a}$ can be
zero and we conclude therefore that $s\in L$ and that

$\phi^{\prime}(a\otimes f)\ =\ \lambda(a)f(s)$ (4.3)

for some $\lambda(a)\geq 0$ and all $f\in C_{\mathbb{R}}(L)$. Every element of
${\operatorname{Aff}}(\Delta)$ is the difference between two positive elements
and it follows therefore that for all $a\in{\operatorname{Aff}}(\Delta)$ there
is a real number $\lambda(a)$ such that (4.3) holds for all $f\in
C_{\mathbb{R}}(L)$. The resulting map $a\mapsto\lambda(a)$ is clearly linear
and positive, and $\lambda(1)=1$ since $1=\phi([[u]])=\phi^{\prime}(1)$. This
implies that there is an $x_{0}\in\Delta$ such that $\lambda(a)=a(x_{0})$ for
all $a\in{\operatorname{Aff}}(\Delta)$. The elements of $\left\\{a\otimes f:\
a\in{\operatorname{Aff}}(\Delta),\ f\in C_{\mathbb{R}}(L)\right\\}$ span a
dense set in $A(\Delta\times L)$ and we find therefore that

$\phi^{\prime}(h)\ =\ h(x_{0},s)\ \ \forall h\in A(\Delta\times L)\ .$ (4.4)

To see that $x_{0}\in F$ when $s\neq 1$, assume for a contradiction that
$x_{0}\notin F$. By Lemma 4.3 we can find $b\in{\operatorname{Aff}}(\Delta)$
such that $b|_{F}=0$ and $b(x_{0})>0$, and since $s\neq 1$ we can find $f\in
C_{\mathbb{R}}(L)$ such that $f(1)=0$ and $f(s)=1$. The function $b\otimes f$
vanishes on $(F\times L)\cup(\Delta\times\\{1\\})$. By Lemma 4.2 and the
definition of $G^{+}$ this implies that $\phi^{\prime}(b\otimes f)=0$, which
contradicts (4.4) since $b(x_{0})f(s)>0$. Hence $x_{0}\in F$ when $s\neq 1$.
It follows that

$\phi(g)\ =\ \phi^{\prime}(\Sigma(g))\ =\
\sum_{m\in\mathbb{Z}}\theta(g_{m})(x_{0})s^{m}\ ,$

and that $x_{0}\in F$ when $s\neq 1$. ∎

###  4.3. The Elliott invariants of $pCp$ and $A$ are isomorphic

######  Lemma 4.9.

Set

$G^{++}=\\{0\\}\cup\left\\{g\in G:\ \sum_{m\in\mathbb{Z}}\theta(g_{m})(x)\ >\
0\ \ \forall x\in\Delta\right\\}\ .$

Then $K_{0}(C)^{+}={G^{++}}/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)$
and $S$ takes $K_{0}(C)^{+}$ onto $H^{+}$.

###### Proof.

Let $g=(g_{m})_{m\in\mathbb{Z}}\in G^{++}\backslash\\{0\\}$ and set
$g^{+}=[[\sum_{m\in\mathbb{Z}}g_{m}]]\in G$. Then
$g^{+}-g\in(\operatorname{id}_{G}-\gamma_{*})(G)$ by Lemma 4.6, and since

$\sum_{m\in\mathbb{Z}}\theta(g^{+}_{m})(x)t^{m}=\sum_{m\in\mathbb{Z}}\theta(g_{m})(x)\geq\epsilon\
,$

for all $(x,t)\in\Delta\times L$, where
$\epsilon=\min_{x\in\Delta}\sum_{m\in\mathbb{Z}}\theta(g_{m})(x)>0$, it
follows that $g^{+}\in G^{+}$. This shows that

${G^{++}}/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)\subseteq{G^{+}}/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)\subseteq
K_{0}(C)^{+}\ .$

Let $z\in K_{0}(C)^{+}\backslash\\{0\\}\subseteq
G/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)$ and choose $g\in G$ such
that $q(g)=z$. Let $x\in\Delta$. The trace $\tau_{x}$ from Lemma 4.8 is
$\gamma$-invariant and there is therefore a trace $\tau_{x}^{\prime}$ on $C$
such that $\tau_{x}^{\prime}|_{B}=\tau_{x}$. Then
${\tau_{x}^{\prime}}_{*}(z)>0$ since $z>0$ in $K_{0}(C)$ and $C$ is simple.
Hence

$0<{\tau_{x}^{\prime}}_{*}(z)={\tau_{x}}_{*}(g)=\
\sum_{m\in\mathbb{Z}}\theta(g_{m})(x)\ .$

It follows that $g\in G^{++}$ and
$z\in{G^{++}}/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)$. We conclude
that $K_{0}(C)^{+}={G^{++}}/\left(\operatorname{id}_{G}-\gamma_{*}\right)(G)$
and $S(K_{0}(C)^{+})\subseteq H^{+}$. When $h\in H^{+}\backslash\\{0\\}$, set
$z=[[h]]$. Then $z\in G^{++}$, $q(z)\in K_{0}(C)^{+}$ and $S(q(z))=h$. ∎

######  Lemma 4.10.

Let $p\in B$ be a projection representing $[[u]]\in G=K_{0}(B)$. The Elliott
invariants of $A$ and $p\left(B\rtimes_{\gamma}\mathbb{Z}\right)p$ are
isomorphic.

###### Proof.

Since $K_{1}(A)=K_{1}(pCp)=0$ it suffices to supplement the isomorphism
$S:(K_{0}(C),K_{0}(C)^{+},[[u]])\to(H,H^{+},u)$ of partially ordered groups
with order unit with an affine homeomorphism $T:T(pCp)\to T(A)$ such that

$T(\tau)_{*}(S(z))=\tau_{*}(z)\ \ \ \forall(\tau,z)\in T(pCp)\times K_{0}(C)\
.$ (4.5)

Let $\tau\in T(pCp)$. By Proposition 4.7 in [CP] there is a unique trace
$\tau^{\prime}$ on $C$ such that $\tau^{\prime}|_{pCp}=\tau$. Then
$\tau^{\prime}|_{B}$ is a $\gamma$-invariant trace on $B$ such that
$\tau^{\prime}(p)=1$. It follows from Lemma 5.10 that there is an $x\in\Delta$
such that ${(\tau^{\prime}|_{B})}_{*}={\tau_{x}}_{*}$ on $K_{0}(B)$. Since $x$
is uniquely determined by $\tau$ we can define $T:T(pCp)\to T(A)$ such that
$T(\tau)=\operatorname{tr}_{x}$. Let $e\in K_{0}(A),\ 0\leq e\leq[1]$. Then
$0\leq[[e]]\leq[[u]]$ in $G$ and hence $[[e]]$ is represented in $K_{0}(B)$ by
a projection $e^{\prime}$ from $pBp\subseteq pCp$. We find that

${\operatorname{tr}_{x}}_{*}(e)=\theta(e)(x)={\tau_{x}}_{*}([[e]])={(\tau^{\prime}|_{B})}_{*}([[e]])=\tau(e^{\prime})\
;$

an equality which shows that $T$ is affine and continuous. To see that $T$ is
surjective let $x\in\Delta$. The trace $\tau_{x}$ from Lemma 4.7 is
$\gamma$-invariant and defines therefore a trace $\tau^{\prime}$ on $C$ such
that $\tau^{\prime}|_{B}=\tau_{x}$. Then
$\tau^{\prime}(p)={\tau_{x}}_{*}([[u]])=1$ and hence $\tau^{\prime}|_{pCp}\in
T(pCp)$ and $T\left(\tau^{\prime}|_{pCp}\right)=\operatorname{tr}_{x}$. To see
that $T$ is also injective consider to traces $\tau_{i},i=1,2$, in $T(pCp)$
and assume that $T(\tau_{1})=T(\tau_{2})$. Then
${(\tau^{\prime}_{1}|_{B})}_{*}={(\tau^{\prime}_{2}|_{B})}_{*}$ on $G$ and
hence $\tau^{\prime}_{1}|_{B}=\tau^{\prime}_{2}|_{B}$ by Lemma 3.5. It follows
from the first of the additional properties 4.5 that
$\tau^{\prime}_{1}=\tau^{\prime}_{2}$ and hence also that
$\tau_{1}={\tau^{\prime}_{1}}|_{pCp}={\tau^{\prime}_{2}}|_{pCp}=\tau_{2}$. We
conclude that $T$ is an affine homeomorphism. Consider $\tau\in T(pCp)$ and
$e\in K_{0}(A),\ 0\leq e\leq 1$. Then $T(\tau)=\operatorname{tr}_{x}$ for some
$x\in\Delta$ and, as observed above, $[[e]]\in G$ is represented by a
projection $e^{\prime}\in pBp$ such that
${\operatorname{tr}_{x}}_{*}(e)=\tau(e^{\prime})$. Hence

$\displaystyle
T(\tau)_{*}\left(S(S^{-1}(e))\right)={\operatorname{tr}_{x}}_{*}(e)=\tau(e^{\prime})=\tau_{*}(q([[e]])=\tau_{*}\left(S^{-1}(e)\right)\
.$

Since the elements of $\left\\{S^{-1}(e):\ e\in K_{0}(A),\ 0\leq
e\leq[1]\right\\}$ generate $K_{0}(pCp)$ as a group, we conclude that (4.5)
holds. ∎

###  4.4. Conclusion

######  Lemma 4.11.

$A$ is $*$-isomorphic to $p(B\rtimes_{\gamma}\mathbb{Z})p$.

###### Proof.

By Lemma 4.10 it remains only to show that $A$ and $pCp$ belong to a class of
simple $C^{*}$-algebras for which the Elliott invariant is complete. For this
we appeal to Corollary D of [CETWW] and we must therefore check that both
algebras are separable, unital, nuclear, $\mathcal{Z}$-stable and satisfy the
UCT. Only $\mathcal{Z}$-stability is not well-known. For $A$ this follows from
Theorem A and Corollary C in [CETWW] and for $pCp$ it follows from the third
of the Additional properties 4.5 combined with Corollary 3.2 in [TW]. ∎

######  Remark 4.12.

Assume that $A$ is UHF. Then the algebras $pCp$ and $A$ occurring in the last
proof only have one trace state and there is an alternative way to deduce the
isomorphism $A\simeq pCp$. The path through the litterature takes more space
to explain, but presents presumably a shorter argument: By Corollary 6.2 in
[MS] it suffices to show that both algebras are unital, separable, simple,
infinite dimensional, nuclear, quasi-diagonal, satisfy the UCT and have strict
comparison. Many of these properties are well-known for both algebras. What
remains is to explain why they are quasi-diagonal and have strict comparison.
Since $pCp$ and $A$ are both exact, simple and unital it follows from
Corollary 4.6 in [Ro] that $\mathcal{Z}$-stability implies strict comparison
and hence it suffices to argue that both algebras are quasi-diagonal and
$\mathcal{Z}$-absorbing. Concerning quasi-diagonality it is well-known that AF
algebras have this property and that it is inherited by subalgebras, so a
straightforward application of [Br] shows that both algebras are quasi-
diagonal. $C$ is $\mathcal{Z}$-absorbing by the third of the additional
properties 4.5 and hence so is $pCp$ by Corollary 3.2 in [TW] since $pCp$ is
stably isomorphic $C$ by [B]. Finally, since $A$ is approximately divisible it
is also $\mathcal{Z}$-absorbing by Theorem 2.3 in [TW].

######  Remark 4.13.

We haven’t used the second of the Additional properties 4.5. It was added
because stability of $B\rtimes_{\gamma}\mathbb{Z}$ combined with Lemma 4.11
and [B] implies that $B\rtimes_{\gamma}\mathbb{Z}\simeq A\otimes\mathbb{K}$; a
fact which is nice to know, but it is not needed for the proof of Theorem 2.1.
111It has been pointed out to me that stability of
$B\rtimes_{\gamma}\mathbb{Z}$ also follows from [HR].

The dual action on $B\rtimes_{\gamma}\mathbb{Z}$ defines by restriction a
$2\pi$-periodic flow on $p(B\rtimes_{\gamma}\mathbb{Z})p$ which we denote by
$\widehat{\gamma}$.

######  Lemma 4.14.

For $\beta\in\mathbb{R}$ there is a $\beta$-KMS state for $\widehat{\gamma}$
on $p(B\rtimes_{\gamma}\mathbb{Z})p$ if and only if $\beta\in K$. For
$\beta\in K\backslash\\{0\\}$ the simplex of $\beta$-KMS states for
$\widehat{\gamma}$ is affinely homeomorphic to $F$ and for $\beta=0$ it
consists of all trace states on $p(B\rtimes_{\gamma}\mathbb{Z})p$ and hence is
affinely homeomorphic to $\Delta$.

###### Proof.

Let $\beta\in K\backslash\\{0\\}$ and $y\in F$. It follows from Lemma 3.1 that
we can define a $\beta$-KMS state on $p(B\times_{\gamma}\mathbb{Z})p$ by
$\tau_{y,e^{\beta}}\circ P|_{p(B\rtimes_{\gamma}\mathbb{Z})p}$. The resulting
map, from $F$ to the simplex of $\beta$-KMS states for $\widehat{\gamma}$ on
$p(B\times_{\gamma}\mathbb{Z})p$, is clearly continuous, affine and injective.
To see that it is surjective, let $\omega$ be a $\beta$-KMS state for
$\widehat{\gamma}$. From Theorem 2.4 in [Th2], or more precisely from the part
of that theorem which follows from Remark 3.3 in [LN], combined with Lemma
3.1, it follows that there is trace $\tau$ on $B$ such that
$\tau\circ\gamma=e^{-\beta}\tau$, $\tau(p)=1$ and $\omega=\tau\circ
P|_{p(B\rtimes_{\gamma}\mathbb{Z})p}$. It follows from Lemma 5.10 that
$\tau_{*}={\tau_{y,e^{\beta}}}_{*}$ on $G$ for some $y\in F$ and then from
Lemma 3.5 that $\tau=\tau_{y,e^{\beta}}$. We conclude that $\omega$ is the
image of $y\in F$ under the map we consider. This shows that the map is an
affine homeomorphism. The same argument, using the traces $\tau_{x}$ from
Lemma 4.7, shows that the simplex of $0$-KMS states for $\widehat{\gamma}$ is
affinely homeomorphic to $\Delta$.

It remains to show that there are no $\beta$-KMS states for $\widehat{\gamma}$
unless $\beta\in K$. The argument for this is almost identical to one we have
given above: Assume that there is $\beta$-KMS state for $\widehat{\gamma}$. It
follow from Remark 3.3 in [LN] that this $\beta$-KMS state is the restriction
to $p(B\rtimes_{\gamma}\mathbb{Z})p$ of a $\beta$-KMS weight for the dual
action on $B\rtimes_{\gamma}\mathbb{Z}$ and then from Lemma 3.1 that there is
a trace $\tau$ on $B$ such that $\tau\circ\gamma=e^{-\beta}\tau$ and
$\tau(p)=1$. Then $\tau_{*}:G\to\mathbb{R}$ is a positive homomorphism such
that $\tau_{*}\circ\gamma_{*}=e^{-\beta}\tau_{*}$ and $\tau_{*}([[u]])=1$. It
follows from Lemma 5.10 that $\beta\in K$. ∎

Now Theorem 2.1, with $K$ compact, follows from Lemma 4.14 and Lemma 4.11.

##  5\. Proof of Theorem 2.1 when $K$ is unbounded

Let $K$ be the closed lower bounded set of real numbers from Theorem 2.1 and
set $L=\left\\{e^{\beta}:\ \beta\in K\right\\}$; a closed subset of
$]0,\infty[$ bounded away from $0$. We have already covered the case where $K$
is also bounded above, so here we assume that $K$ and also $L$ is unbounded.

###  5.1. The second construction

Let $P_{\Delta}(t)$ denote the vector space of Laurent polynomials in $t$ with
coefficients from ${\operatorname{Aff}}(\Delta)$. Thus $P_{\Delta}(t)$
consists of the functions $f:\Delta\times]0,\infty[\to\mathbb{R}$ of the form

$f(x,t)=\sum_{n=-\infty}^{\infty}a_{n}(x)t^{n}\ ,$ (5.1)

where $a_{n}\in{\operatorname{Aff}}(\Delta)$ for all $n$ and only finitely
many $a_{n}$’s are non-zero. Given $f\in P_{\Delta}(t)$ as in (5.1) we set

$\mathbb{L}(f)=\max\left\\{n\in\mathbb{Z}:\ a_{n}\neq 0\right\\}\ $

when $f\neq 0$, and let $\mathbb{L}(0)=-\infty$. We will refer to
$\mathbb{L}(f)$ as the degree of $f$ and the element
$a_{\mathbb{L}(f)}\in{\operatorname{Aff}}(\Delta)$ as the leading coefficient
of $f$.

Let $F$ be a closed non-empty face in $\Delta$. For $f\in P_{\Delta}(t)$ we
write $0\prec f$ when there is an $\epsilon>0$ such that

$t^{-\mathbb{L}(f)}f(x,t)\geq\epsilon\ \ \ \ \forall(x,t)\in(F\times
L)\cup(\Delta\times\\{1\\})\ .$

Thus $0\prec f$ if and only if $f$ is strictly positive on $(F\times
L)\cup(\Delta\times\\{1\\})$ and the leading coefficient of $f$ is strictly
positive on $\Delta$. Given $f,g\in P_{\Delta}(t)$, we write $f\prec g$ when
$0\prec g-f$. As in Section 4.2 we set $G=\oplus_{\mathbb{Z}}H$ and define a
homomorphism $\Sigma:G\to P_{\Delta}(t)$ such that

$\Sigma(g)(x,t)=\sum_{m\in\mathbb{Z}}\theta(g_{m})(x)t^{m}\ .$

Set

$G^{+}=\left\\{g\in G:\ 0\prec\Sigma(g)\right\\}\cup\\{0\\}\ .$

We aim now to show that $(G,G^{+})$ is a Riesz group, and hence also a
dimension group in the sense of [EHS]. This means that we must prove that

  * •

$G^{+}+G^{+}\subseteq G^{+}$,

  * •

$G^{+}\cap(-G^{+})=\\{0\\}$,

  * •

$g\in G,\ n\in\mathbb{N}\backslash\\{0\\},\ ng\in G^{+}\ \Rightarrow\ g\in
G^{+}$,

  * •

$G^{+}-G^{+}=G$, and

  * •

$G$ has the Riesz interpolation property.

The first three items are easily checked. To show that $G^{+}-G^{+}=G$, let
$g\in G$. Choose $l\in\mathbb{N}$ bigger than any $n\in\mathbb{N}$ for which
$\theta(g_{n})$ is non-zero and set $l=0$ if $\theta(g_{n})=0$ for all $n$.
Choose $N\in\mathbb{N}$ such that
$Nt^{l}>\sum_{n\in\mathbb{Z}}\left|\theta(g_{n})(x)\right|t^{n}$ for all
$n\in\mathbb{N}$ and $(x,t)\in\Delta\times L$. Define $v\in G$ such that

$v_{n}=\begin{cases}0,\ n\neq l\ ,\\\ Nu,\ n=l\ .\end{cases}$

Then $v,v-g\in G^{+}$ and $g=v-(v-g)$.

It remains to show that $G$ has the Riesz interpolation property. The next
subsection is devoted to that.

####  5.1.1. $(G,G^{+})$ is a Riesz group

For $g\in G$, set

$\mathbb{L}(g)=\max\\{n\in\mathbb{Z}:g_{n}\neq 0\\}$

when $g\neq 0$, and $\mathbb{L}(0)=-\infty$. Then
$\mathbb{L}(\Sigma(g))\leq\mathbb{L}(g)$.

######  Lemma 5.1.

Let $a\in{\operatorname{Aff}}(\Delta)$ and let $f:L\to\mathbb{R}$ be a
continuous function such that $\lim_{t\to\infty}f(t)=0$. Let $\epsilon>0$ and
$N\in\mathbb{N}$. There is a $g\in G$ such that $\mathbb{L}(g)\leq-N$ and

$\left|\Sigma(g)(x,t)-f(t)a(x)\right|\leq\epsilon\ \
\forall(x,t)\in\Delta\times L\ .$

###### Proof.

By the Stone-Weierstrass theorem there is a real Laurent polynomial
$Q(t)=\sum_{n=-M}^{-N}c_{n}t^{n}$ of degree at most $-N$ such that

$\left|a(x)\right|\left|Q(t)-f(t)\right|\leq\frac{\epsilon}{2}$

for all $x\in\Delta$ and all $t\in L$. Since $\theta(H)$ is dense in
${\operatorname{Aff}}(\Delta)$ there are elements $h_{n}\in H$ such that

$\left|\theta(h_{n})(x)-c_{n}a(x)\right|t^{n}\leq\frac{\epsilon}{2(M-N)}$

for all $(x,t)\in\Delta\times L$ and all $n\in\\{-M,-M+1,\cdots,-N\\}$. Define
$g\in G$ such that

$g_{n}=\begin{cases}0\ ,\ &n\leq-M-1\ \text{or}\ n\geq-N+1\ ,\\\ h_{n}\ ,\ &\
-M\leq n\leq-N\ .\end{cases}$

Then $\mathbb{L}(g)\leq-N$ and

$\displaystyle\left|\Sigma(g)(x,t)-f(t)a(x)\right|$
$\displaystyle\leq\left|\sum_{n=-M}^{-N}\theta(h_{n})(x)t^{n}-\sum_{n=-M}^{-N}c_{n}a(x)t^{n}\right|+\left|Q(t)a(x)-f(t)a(x)\right|\leq\epsilon\
.$

∎

######  Lemma 5.2.

Let $g\in G$ and let $F\subseteq\mathbb{Z}$ be a finite set. Assume that
$g_{n}=0,\ |n|\geq J\geq 1$. Let $N\in\mathbb{N}$ such that $J-N-d<-J$ for all
$d\in F\cup\\{0\\}$. Let $q$ be a real-valued Laurent polynomial of degree at
most $-N$. For any $\epsilon>0$ there is an element $g^{\prime}\in G$ such
that $g^{\prime}_{n}=g_{n}$ when $n\geq-J$, and

$\left|t^{-d}(1+q(t))\Sigma(g)(x,t)-t^{-d}\Sigma(g^{\prime})(x,t)\right|\leq\epsilon$

for all $d\in F$ and all $(x,t)\in\Delta\times L$.

###### Proof.

By assumption we can write

$q(t)\Sigma(g)(x,t)=\sum_{n=B}^{A}a_{n}(x)t^{n}\ ,$

where $a_{n}\in{\operatorname{Aff}}(\Delta)$ for all $n$ and $B<A\leq J-N$.
Since $\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ and since
$J-N-d<-J<0$ we can find $h_{n}\in H,\ B\leq n\leq A$, such that

$\left|t^{-d}q(t)\sum_{n=B}^{A}a_{n}(x)t^{n}-t^{-d}\sum_{n=B}^{A}\theta(h_{n})(x)t^{n}\right|\leq\epsilon$

for all $d\in F$ and all $(x,t)\in\Delta\times L$. Define $g^{\prime}\in G$
such that $g^{\prime}_{n}=h_{n},\ B\leq n\leq A$, and $g^{\prime}_{n}=g_{n}$
for $n\notin[B,A]$. Then

$\displaystyle\left|t^{-d}(1+q(t))\Sigma(g)(x,t)-t^{-d}\Sigma(g^{\prime})(x,t)\right|$
$\displaystyle=\left|t^{-d}q(t)\sum_{n=B}^{A}a_{n}(x)t^{n}-t^{-d}\sum_{n=B}^{A}\theta(h_{n})(x)t^{n}\right|\leq\epsilon\
$

for all $d\in F$ and all $(x,t)\in\Delta\times L$. ∎

######  Lemma 5.3.

Let $F:\Delta\times L\to\mathbb{R}$ be a continuous function such that
$\Delta\ni x\mapsto F(x,t)$ is affine for all $t\in L$ and
$\lim_{t\to\infty}F(x,t)=0$ uniformly in $x$. Let $J\in\mathbb{N}$. For every
$\delta>0$ there is a $g\in G$ such that $\mathbb{L}(g)\leq-J$, and

$\left|\Sigma(g)(x,t)-F(x,t)\right|\leq\delta$

for all $(x,t)\in\Delta\times L$.

###### Proof.

Let $\kappa>0$. There is then an $R>0$ such that
$\left|F(x,t)\right|\leq\kappa$ for all $x\in\Delta$ and all $t\geq R$. For
each $t^{\prime}\in L\cap\left]0,R\right]$, there is an element
$h_{t^{\prime}}\in H$ such that

$\left|F(x,t)-\theta(h_{t^{\prime}})(x)\right|<\kappa\ $

for all $x\in\Delta$ and all $t$ in an open neighborhood of $t^{\prime}$. By
compactness of $L\cap\left]0,R\right]$, we can therefore find open sets
$U_{i}$ in $\mathbb{R}$ and elements $h_{i}\in H,i=1,2,\cdots,n$, such that

$\left|F(x,t)-\theta(h_{i})(x)\right|\leq\kappa$

for all $x\in\Delta\times(L\cap U_{i}),\ i=1,2,\cdots,n$, and
$L\cap\left]0,R\right]\subseteq\bigcup_{i=1}^{n}U_{i}$. Let
$\varphi_{i}:\mathbb{R}\to\mathbb{R},\ i=1,2,\cdots,n$, be continuous non-
negative compactly supported functions such that
$\operatorname{supp}\varphi_{i}\subseteq U_{i}$ for all $i$,
$\sum_{i=1}^{n}\varphi_{i}\leq 1$ and $\sum_{i=1}^{n}\varphi_{i}(x)=1$ for all
$x\in L\cap\left]0,R\right]$. By Lemma 5.1 there are elements $g_{i}\in G$
such that $\mathbb{L}(g_{i})\leq-J$ and

$\left|\Sigma(g_{i})(x,t)-\varphi_{i}(t)\theta(h_{i})(x)\right|\leq\frac{\kappa}{n}$

for all $(x,t)\in\Delta\times L$. Set $g=\sum_{i=1}^{n}g_{i}$ and note that
$\mathbb{L}(g)\leq-J$. Furthermore,

$\displaystyle\left|\Sigma(g)(x,t)-F(x,t)\right|$
$\displaystyle\leq\left|(1-\sum_{i=1}^{n}\varphi_{i}(t))F(x,t)\right|+\sum_{i=1}^{n}\left|\Sigma(g_{i})(x,t)-\varphi_{i}(t)F(x,t)\right|$
$\displaystyle\leq\kappa+\sum_{i=1}^{n}\left(\left|\Sigma(g_{i})(x,t)-\varphi_{i}(t)\theta(h_{i})(x)\right|+\varphi_{i}(t)\left|\theta(h_{i})(x)-F(x,t)\right|\right)$
$\displaystyle\leq 3\kappa$

for all $(x,t)\in\Delta\times L$. ∎

######  Lemma 5.4.

$(G,G^{+})$ is a dimension group.

###### Proof.

As observed above it remains only to establish the Riesz interpolation
property. Let $u^{i},v^{j}\in G,\ i,j\in\\{1,2\\}$, satisfy that $u^{i}\leq
v^{j}$ in $G$ for all $i,j$. We must find $g^{\prime}\in G$ such that
$u^{i}\leq g^{\prime}\leq v^{j}$ for all $i,j$. If
$u^{i^{\prime}}=v^{j^{\prime}}$ for some $i^{\prime},j^{\prime}\in\\{1,2\\}$,
set $g^{\prime}=v^{j^{\prime}}$. Then $u^{i}\leq g^{\prime}\leq v^{j}$ for all
$i,j$ and we are done. Assume therefore that $\Sigma(u^{i})\prec\Sigma(v^{j})$
for all $i,j\in\\{1,2\\}$.

######  Observation 5.5.

There is a $g\in G$, an $R>1$ and an $\epsilon>0$, such that

$t^{-l_{i}}(\Sigma(g)(x,t)-\Sigma(u^{i})(x,t))>\epsilon$

for all $(x,t)\in F\times(L\cap[R,\infty))$, $i=1,2$, where

$l_{i}=\mathbb{L}(\Sigma(g)-\Sigma(u^{i}))\ ,$

and such that

$t^{-l^{\prime}_{j}}(\Sigma(v^{j})(x,t)-\Sigma(g)(x,t))>\epsilon$

for all $(x,t)\in F\times(L\cap[R,\infty))$, $j=1,2$, where

$l^{\prime}_{j}=\mathbb{L}(\Sigma(v^{j})-\Sigma(g))\ .$

###### Proof.

Choose $N\in\mathbb{N}$ such that $u^{i}_{k}=v^{j}_{k}=0$ for all
$i,j\in\\{1,2\\}$ when $|k|\geq N$. Define
$\overline{u}^{i}\in\oplus_{k=0}^{\infty}\theta(H)$ such that

$\overline{u}^{i}_{k}=\theta(u^{i}_{N-k})\ ,$

and $\overline{v}^{j}\in\oplus_{k=0}^{\infty}\theta(H)$ such that

$\overline{v}^{j}_{k}=\theta(v^{j}_{N-k})\ .$

We consider here $\theta(H)$ is a partially ordered group in the order
inherited from the strict order of ${\operatorname{Aff}}(F)$; i.e.
$\theta(H)^{+}$ consists of $0$ and the elements $f\in\theta(H)$ such that
$f(x)>0$ for all $x\in F$. It is well-known that $\theta(H)$ then has the
Riesz interpolation property, cf. Lemma 3.1 and Lemma 3.2 in [EHS]. The direct
sum $\oplus_{k=0}^{\infty}\theta(H)$, in turn, is given the corresponding
lexicographic order $<_{lex}$, where
$(h_{k})_{k=0}^{\infty}<_{lex}(h^{\prime}_{k})_{k=0}^{\infty}$ means that
$h_{k}=h^{\prime}_{k}$ for all $k$, or $h_{k_{0}}<h^{\prime}_{k_{0}}$ in
$\theta(H)$ where $k_{0}=\min\\{k:\ h_{k}\neq h^{\prime}_{k}\\}$. Since
$\Sigma(u^{i})\prec\Sigma(v^{j})$ for all $i,j\in\\{1,2\\}$ we have that

$\overline{u}^{i}<_{lex}\overline{v}^{j}$

for all $i,j\in\\{1,2\\}$. By Theorem 3.10 in [E2] the group
$\oplus_{k=0}^{\infty}\theta(H)$ has the Riesz interpolation property in the
lexicographic order222Theorem 3.10 in [E2] is very general and the argument
concerning Riesz groups uses the decomposition property. However, a major step
in the proof of Lemma 3.2 in [BEK] gives the arguments directly for the
interpolation property, albeit in the colexicographic order. and there is
therefore an element $h^{\prime\prime}\in\oplus_{k=0}^{\infty}\theta(H)$ such
that $\overline{u}^{i}<_{lex}h^{\prime\prime}<_{lex}\overline{v}^{j}$ for all
$i,j\in\\{1,2\\}$. If
$h^{\prime\prime}\notin\\{\overline{u}^{1},\overline{u}^{2},\overline{v}^{1},\overline{v}^{2}\\}$
we define $g\in G$ such that $g_{k}=0$ when $k>N$ and such that
$\theta(g_{k})=h^{\prime\prime}_{N-k}$ when $k\leq N$, and $g$ will then have
the desired properties because the leading coefficients of
$\Sigma(g)-\Sigma(u^{i})$ and $\Sigma(v^{j})-\Sigma(g),\ i,j=1,2$, will all be
strictly positive. If $h^{\prime\prime}=\overline{u}^{1}$ it follows that
$\overline{u}^{2}<_{lex}\overline{u}^{1}$ which implies that
$\Sigma(u^{2})=\Sigma(u^{1})$ or that there is an $R^{\prime}>1$ and a
$\delta>0$ such that

$t^{-l}(\Sigma(u^{1})(x,t)-\Sigma(u^{2})(x,t))\geq\delta>0$

for all $(x,t)\in F\times[R^{\prime},\infty)$ where
$l=\mathbb{L}\left(\Sigma(u^{1})-\Sigma(u^{2})\right)$. In both these cases we
choose, as we can, an element $d\in H$ such that

$0<\theta(d)(x)t^{-N-1}<\frac{1}{2}(\Sigma(v^{j})(x,t)-\Sigma(u^{1})(x,t))$

for all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$, $j=1,2$, and define
$g\in G$ such that

$g_{k}=\begin{cases}u^{1}_{k},&\ -N\leq k\leq N,\\\ d,&\ k=-N-1,\\\ 0,&\
k\notin[-N-1,N]\ .\end{cases}\ $

Then $g$ will have the desired property. The cases where
$h^{\prime\prime}\in\\{\overline{u}^{2},\overline{v}^{1},\overline{v}^{2}\\}$
are handled in a similar way. ∎

We fix now $g$, $R$ and $\epsilon$ as in Observation 5.5 and choose
$J\in\mathbb{N}$ such that $g^{\prime}_{n}=0$ when $|n|>J$ and
$g^{\prime}\in\left\\{g,v^{1},v^{2},u^{1},u^{2}\right\\}$. And we choose
$J^{\prime}\in\mathbb{N}$ such that

$J^{\prime}>2J-d$

for all $d\in\\{0,l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$.

######  Observation 5.6.

There is an element $g^{(1)}\in G$ and an $\epsilon_{1}>0$ such that
$\mathbb{L}(g^{(1)})\leq-J^{\prime}$ and

$t^{-d}\Sigma(u^{i})(x,t)+\epsilon_{1}<t^{-d}\Sigma(g^{(1)})(x,t)<t^{-d}\Sigma(v^{j})(x,t)-\epsilon_{1}$

for all $(x,t)\in(F\times L\cap]0,R+1])\cup(\Delta\times\\{1\\})$ and all
$d\in\\{l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$.

###### Proof.

Since $\Sigma(u^{i})(x,t)<\Sigma(v^{j})(x,t)$ for all $(x,t)$ in the compact
set $(F\times L\cap]0,R+1])\cup(\Delta\times\\{1\\})$ there is a $\delta>0$
such that

$t^{-d}\Sigma(u^{i})(x,t)+\delta<t^{-d}\Sigma(v^{j})(x,t)-\delta$

for all $(x,t)\in(F\times L\cap]0,R+1])\cup(\Delta\times\\{1\\})$ and all
$d\in\\{l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$. By using the Riesz
interpolation property of ${\operatorname{Aff}}(F)$ and
${\operatorname{Aff}}(\Delta)$ as is in the proof of Lemma 4.4 we construct a
continuous function

$a:\Delta\times(L\cap]0,R+1])\to\mathbb{R}$

such that $x\mapsto a(x,t)$ is affine for all $t\in L\cap]0,R+1]$ and

$t^{-d}\Sigma(u^{i})(x,t)+\delta<t^{-d}a(x,t)<t^{-d}\Sigma(v^{j})(x,t)-\delta$

for all $(x,t)\in(F\times L\cap]0,R+1])\cup(\Delta\times\\{1\\})$ and all
$d\in\\{l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$. Let $r$ be the largest
element in $L\cap]0,R+1]$ and extend $a$ to $\Delta\times L$ such that

$a(x,t)=\begin{cases}(r+1-t)a(x,r),\ &t\in[r,r+1]\ ,\\\ 0,\ &t\geq r+1\
.\end{cases}$

It follows from Lemma 5.3 that there is an element $g^{(1)}\in G$ such that
$\mathbb{L}(g^{(1)})\leq-J^{\prime}$ and

$\left|t^{-d}\Sigma(g^{(1)})(x,t)-t^{-d}a(x,t)\right|\leq\frac{\delta}{2}$

for all $(x,t)\in(F\times L\cap]0,R+1])\cup(\Delta\times\\{1\\})$ and all
$d\in\\{l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$. Set
$\epsilon_{1}=\frac{\delta}{2}$. ∎

Let $\psi:]0,\infty[\to[0,1]$ be the continuous function such that
$\psi(t)=1,t\leq R$, $\psi(t)=0,\ t\geq R+1$, and $\psi$ is linear on
$[R,R+1]$. Choose $\kappa>0$ such that

$4\kappa<\min\\{\epsilon,\epsilon_{1}\\}\ $

where $\epsilon>0$ comes from Observation 5.5 and $\epsilon_{1}>0$ from
Observation 5.6. Find a real-valued Laurent polynomial $q$ of degree no more
than $-J^{\prime}$ such that

$\left|t^{-d}\psi(t)\Sigma(g^{(1)})(x,t)-t^{-d}q(t)\Sigma(g^{(1)})(x,t)\right|\leq\kappa$
(5.2)

for all $(x,t)\in\Delta\times L$ and all
$d\in\\{l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$. This is possible
because $\psi|_{L}\in C_{0}(L)$, so that $\psi|_{L}$ can be approximated by
Laurent polynomials of degree as most $-J^{\prime}$, and

$\sup_{(x,t)\in\Delta\times L}\left|t^{-d}\Sigma(g^{(1)})(x,t)\right|<\infty\
$

since $\mathbb{L}(g^{(1)})-d\leq 0$. Since

$\sup_{(x,t)\in\Delta\times
L}t^{-l_{i}}\left(\Sigma(g)(x,t)-\Sigma(u^{i})(x,t)\right)<\infty$

and

$\sup_{(x,t)\in\Delta\times
L}t^{-l^{\prime}_{j}}\left(\Sigma(v^{j})(x,t)-\Sigma(g)(x,t)\right)<\infty$

for all $i,j$, we can also arrange that

$\left|(q(t)-\psi(t))t^{-l_{i}}\left(\Sigma(g)(x,t)-\Sigma(u^{i})(x,t)\right)\right|<\kappa$
(5.3)

and

$\left|(q(t)-\psi(t))t^{-l^{\prime}_{j}}\left(\Sigma(v^{j})(x,t)-\Sigma(g)(x,t)\right)\right|<\kappa$
(5.4)

for all $(x,t)\in\Delta\times L$ and all $i,j$.

We apply now Lemma 5.2 to get an element $g^{(2)}\in G$ such that
$g^{(2)}_{n}=g_{n}$ for all $-J\leq n\leq J$, $g^{(2)}_{n}=0,\ n\geq J+1$, and

$\left|t^{-d}(1-q(t))\Sigma(g)(x,t)-t^{-d}\Sigma(g^{(2)})(x,t)\right|\leq\kappa$
(5.5)

for all $(x,t)\in\Delta\times L$, and all
$d\in\\{0,l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$.

######  Observation 5.7.

There is an element $g^{(3)}\in G$ such that $\mathbb{L}(g^{(3)})<-J$ and

$\left|t^{-d}q(t)\Sigma(g^{(1)})(x,t)-t^{-d}\Sigma(g^{(3)})(x,t)\right|\leq\kappa$
(5.6)

for all $(x,t)\in\Delta\times L$ and all
$d\in\\{l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$.

###### Proof.

Write

$q(t)\Sigma(g^{(1)})(x,t)=\sum_{-M}^{-2J^{\prime}}a_{n}(x)t^{n}$

where $M>2J^{\prime}$. Let $\kappa^{\prime}>0$. Since $\theta(H)$ is dense in
${\operatorname{Aff}}(\Delta)$ we can find $g^{(3)}\in G$ such that
$g^{(3)}_{n}=0$ when $n\notin[-M,-2J^{\prime}]$ and

$\left|\theta(g^{(3)}_{n})(x)-a_{n}(x)\right|\leq\kappa^{\prime}$

for all $x\in\Delta$ and all $-M\leq n\leq-2J^{\prime}$. Then

$\displaystyle\left|t^{-d}q(t)\Sigma(g^{(1)})(x,t)-t^{-d}\Sigma(g^{(3)})(x,t)\right|$
$\displaystyle\leq\kappa^{\prime}(M-2J^{\prime})\sup\left\\{t^{n}:\ -M-d\leq
n\leq-2J^{\prime}-d,\ t\in L\right\\}$

for all $(x,t)\in\Delta\times L$ and all
$d\in\\{l_{1},l_{2},l^{\prime}_{1},l^{\prime}_{2}\\}$. Since
$-2J^{\prime}-d<0$,

$\sup\left\\{t^{n}:\ -M-d\leq n\leq-2J^{\prime}-d,\ t\in L\right\\}$

is finite for all $d$ and we can therefore arrange that (5.6) holds. ∎

Set $g^{(4)}=g^{(3)}+g^{(2)}$. Since $g^{(4)}_{n}=g_{n}$ for all $-J\leq n\leq
J$ and $g^{(4)}_{n}=0,\ n\geq J+1$, we find that

$\mathbb{L}(\Sigma(g^{(4)})-\Sigma(u^{i}))=\mathbb{L}(\Sigma(g)-\Sigma(u^{i}))=l_{i}$

and

$\mathbb{L}(\Sigma(v^{j})-\Sigma(g^{(4)}))=\mathbb{L}(\Sigma(v^{j})-\Sigma(g))=l^{\prime}_{j}$

for all $i,j$. Furthermore, for all $(x,t)\in\Delta\times L$, we find by using
(5.6), (5.5), (5.2) and (5.3), that

$\displaystyle t^{-l_{i}}(\Sigma(g^{(4)})(x,t)-\Sigma(u^{i})(x,t))$
$\displaystyle=t^{-l_{i}}\left(\Sigma(g^{(3)})(x,t)+\Sigma(g^{(2)})(x,t)-\Sigma(u^{i})(x,t)\right)$
$\displaystyle\geq
t^{-l_{i}}(q(t)\Sigma(g^{(1)})(x,t)+(1-q(t))\Sigma(g)(x,t)-\Sigma(u^{i})(x,t))-2\kappa$
$\displaystyle=t^{-l_{i}}q(t)(\Sigma(g^{(1)})(x,t)-\Sigma(u^{i})(x,t))$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+(1-q(t))\left(\Sigma(g)(x,t)-\Sigma(u^{i})(x,t)\right)-2\kappa$
$\displaystyle\geq t^{-l_{i}}\psi(t)(\Sigma(g^{(1)})(x,t)-\Sigma(u^{i})(x,t))$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+(1-q(t))t^{-l_{i}}\left(\Sigma(g)(x,t)-\Sigma(u^{i})(x,t)\right)-3\kappa\ $
$\displaystyle\geq\psi(t)t^{-l_{i}}(\Sigma(g^{(1)})(x,t)-\Sigma(u^{i})(x,t))$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+(1-\psi(t))t^{-l_{i}}\left(\Sigma(g)(x,t)-\Sigma(u^{i})(x,t)\right)-4\kappa\
$

for all $(x,t)\in\Delta\times L$ and $i=1,2$. Now the properties of $g$ and
$g^{(1)}$ stipulated in Observation 5.5 and Observation 5.6, and the
definition of $\psi$, imply that

$\displaystyle\psi(t)t^{-l_{i}}(\Sigma(g^{(1)})(x,t)-\Sigma(u^{i})(x,t))$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+(1-\psi(t))t^{-l_{i}}\left(\Sigma(g)(x,t)-\Sigma(u^{i})(x,t)\right)-4\kappa$
$\displaystyle\geq\min\\{\epsilon,\epsilon_{1}\\}-4\kappa\ $

for all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$ and $i=1,2$.
Essentially identical estimates show that

$t^{-l^{\prime}_{j}}\left(\Sigma(v^{j})(x,t)-\Sigma(g^{(4)})(x,t)\right)\geq\min\\{\epsilon,\epsilon_{1}\\}-4\kappa$

for all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$ and $j=1,2$, and since
$\min\\{\epsilon,\epsilon_{1}\\}-4\kappa>0$ we conclude that $u^{i}\leq
g^{(4)}\leq v^{j}$ in $G$ for all $i,j$. ∎

Recall that an ideal $J$ in $(G,G^{+})$ is a subgroup $J$ of $G$ such that
$J=J\cap G^{+}-J\cap G^{+}$ and $0\leq h\leq g\in J\Rightarrow h\in J$. We
shall not need to identify all ideals in $(G,G^{+})$; it suffices here to
establish the following:

######  Lemma 5.8.

Let $\rho$ be the automorphism (4.1) and let $J$ be a non-zero ideal in
$(G,G^{+})$ such that $\rho(J)=J$. It follows that $J=G$.

###### Proof.

Let $0\neq g\in J\cap G^{+}$. For $i\in\mathbb{Z}$ and $h\in H$ define
$h^{(i)}\in G$ such that $(h^{(i)})_{i}=h$ and $(h^{(i)})_{j}=0,\ j\neq i$.
Let $h\in H^{+}\backslash\\{0\\}$. Then $h^{(i)}\in G^{+}$ for all
$i\in\mathbb{Z}$. Set $i^{\prime}=\mathbb{L}(\Sigma(g))$. Since

$t^{-i^{\prime}}\Sigma(g)(x,t)\geq\epsilon>0$

on $(F\times{L})\cup(\Delta\times\\{1\\})$ for some $\epsilon>0$, there is an
$N\in\mathbb{N}$ such that

$Nt^{-i^{\prime}}\Sigma(g)(x,t)-\theta(h)(x)\geq 1$

for all $(x,t)\in(F\times{L})\cup(\Delta\times\\{1\\})$, which implies that

$0\leq h^{(i^{\prime})}\leq Ng$

in $G^{+}$ and hence that $h^{(i^{\prime})}\in J$. Since $\rho(J)=J$ and
$\rho({h}^{(i)})={h}^{(i-1)}$ for all $i$, it follows that ${h}^{(i)}\in J$
for all $i\in\mathbb{Z}$. The elements $h^{(i)},h\in H^{+},i\in\mathbb{Z}$,
generate $G$ and we conclude therefore that $J=G$. ∎

######  Lemma 5.9.

$(G,G^{+})$ has large denominators.

###### Proof.

We adopt the notation from the proof of Lemma 5.8. Let $g\in
G^{+}\backslash\\{0\\}$ and $n\in\mathbb{N}$ be given. Set
$l=\mathbb{L}(\Sigma(g))$. There is an $\epsilon>0$ such that
$t^{-l}\Sigma(g)(x,t)\geq\epsilon$ for all $(x,t)\in(F\times
L)\cup(\Delta\times\\{1\\})$. Since $\theta(H)$ is dense in
${\operatorname{Aff}}(\Delta)$, there is an element $h\in H$ such that
$0<\theta(h)(x)<\frac{\epsilon}{n}$ for all $x\in\Delta$. Then $0\leq
nh^{(l)}\leq g$ in $G$. On the other hand, there is an $m\in\mathbb{N}$ so big
that

$t^{-l}\Sigma(g)(x,t)+1\leq m\theta(h)(x)$

for all $(x,t)\in\Delta\times L$ and then $g\leq mh^{(l)}$. ∎

####  5.1.2. Completing the proof when $K$ is unbounded

From here the arguments are very similar, and many identical to those from the
compact case.

It follows from Lemma 5.4 and [EHS] that there is an AF algebra $B$ whose
$K_{0}$-group and dimension range is isomorphic to $(G,G^{+})$ and from [E1]
we conclude that $B$ is stable and that there is an automorphism $\gamma$ of
$B$ such that $\gamma_{*}=\rho$ under the identification $K_{0}(B)=G$. By
Lemma 5.9 and Lemma 3.4 we can choose $\gamma$ such that it has the same
additional properties as listed in 4.5. Set $C=B\rtimes_{\gamma}\mathbb{Z}$.
No power of $\gamma$ is inner since $\gamma_{*}^{k}\neq\operatorname{id}_{G}$
for $k\neq 0$ and it follows from Lemma 5.8 that $B$ is $\gamma$-simple, and
then from [Ki1] that $C$ is simple.

The traces $\tau_{y,t}$ and $\tau_{x}$ on $B$ can be defined by the same
formulas as in Lemma 4.7, and the following is an exact copy of Lemma 4.8, but
the different assumptions on $K$ and the different orderings of $G$
necessitate some slight changes to the proof.

######  Lemma 5.10.

Let $\phi:G\to\mathbb{R}$ be a positive homomorphism such that
$\phi([[u]])=1$. Assume that $\phi\circ\gamma_{*}=s^{-1}\phi$ for some
$s\in]0,\infty[$. Then $s\in L$ and when $s\neq 1$ it follows that
$\phi={\tau_{y,s}}_{*}$ for some $y\in F$, and when $s=1$ it follows that
$\phi={\tau_{x}}_{*}$ for some $x\in\Delta$.

###### Proof.

In the following we denote by $C_{0}(\Delta\times L)$ and $C_{0}(L)$ the set
of continuous real-valued functions vanishing at infinity on $\Delta\times L$
and $L$, respectively, and we denote by $\mathcal{A}(\Delta\times L)$ the
elements $f$ of $C_{0}(\Delta\times L)$ with the property that $x\mapsto
f(x,t)$ is affine for all $t\in L$; a closed subspace of $C_{0}(\Delta\times
L)$. Set

$G^{\prime}=\left\\{g\in G:\ \mathbb{L}(g)\leq-1\right\\}=\left\\{g\in G:\
g_{n}=0\ \forall n\geq 0\right\\}\ ;$

a subgroup of $G$. Then $\Sigma(G^{\prime})\subseteq\mathcal{A}(\Delta\times
L)$, and we claim that there is continuous linear map
$\phi^{\prime}:\mathcal{A}(\Delta\times{L})\to\mathbb{R}$ such that
$\phi=\phi^{\prime}\circ\Sigma$ on $G^{\prime}$. For this and later purposes
we establish the following

######  Observation 5.11.

Let $h\in H$ and $r\in\mathbb{R}$ be such that $\theta(h)(x)<r$ for all
$x\in\Delta$. It follows that $\phi([[h]])<r$.

To prove this let $k\in\mathbb{Z}$ and $l\in\mathbb{N}$ be such that
$\theta(h)(x)<\frac{k}{l}\leq r$ for all $x\in\Delta$. Then $l[[h]]\leq
k[[u]]$ in $G$ and hence $l\phi([[h]])\leq k$, from which the conclusion
follows.

Assume $g\in G$ and that $\Sigma(g)=0$. Let $N\in\mathbb{N}$. Since
$\theta(H)$ is dense in ${\operatorname{Aff}}(\Delta)$ there is a $c\in H$
such that $0<\theta(c)(x)<N^{-1}$ for all $x\in\Delta$ and Observation 5.11
implies that $0\leq\theta([[c]])\leq N^{-1}$. Since $\mathbb{L}(\Sigma(\pm
g+[[c]]))=0$ and $\Sigma(\pm g+[[c]])(x,t)\geq\epsilon>0$, where
$\epsilon=\min_{y\in\Delta}\theta(h)(y)$, it follows that $\pm g+[[c]]\in
G^{+}$. We find in this way that $-\frac{1}{N}\leq\phi(g)\leq\frac{1}{N}$.
Letting $N\to\infty$ we conclude that $\phi(g)=0$, and it follows that there
is a homomorphism $\phi^{\prime}:\Sigma(G^{\prime})\to\mathbb{R}$ such that
$\phi^{\prime}\circ\Sigma=\phi$. Let $f\in\Sigma(G^{\prime})$; say
$f=\Sigma(g)$, where $g\in G^{\prime}$, and let $k,l\in\mathbb{N}$ be natural
numbers such that $|f(x,t)|<\frac{k}{l}$ for all $(x,t)\in\Delta\times{L}$.
Since $\mathbb{L}\left(\Sigma(k[[u]]\pm lg)\right)=0$ this implies that
$k[[u]]\pm lg\in G^{+}$ and hence

$0\ \leq\ \phi(k[[u]]\pm lg)\ =\ k\pm l\phi^{\prime}(f)\ .$

It follows that $\left|\phi^{\prime}(f)\right|\leq\frac{k}{l}$, proving that
$\phi^{\prime}$ is Lipshitz continuous on $\Sigma(G^{\prime})$. Since
$\Sigma(G^{\prime})$ is dense in $\mathcal{A}(\Delta\times{L})$ by Lemma 5.3
it follows that $\phi^{\prime}$ extends by continuity to a linear map
$\phi^{\prime}:\mathcal{A}(\Delta\times{L})\to\mathbb{R}$ such that
$\phi^{\prime}\circ\Sigma(g)=\phi(g)$ for $g\in G^{\prime}$.

Let $T:\mathcal{A}(\Delta\times{L})\to\mathcal{A}(\Delta\times{L})$ be the
operator

$T(\psi)(x,t)=t^{-1}\psi(x,t)\ .$

The equality

$s^{-1}\phi^{\prime}=\phi^{\prime}\circ T\ $ (5.7)

is established in the same way as in the proof of Lemma 4.8. When
$a\in{\operatorname{Aff}}(\Delta)$ and $f\in C_{0}({L})$ we denote by
$a\otimes f$ the function $\Delta\times{L}\ni(x,s)\mapsto a(x)f(s)$. Assume
$a(x)f(t)\geq 0$ for all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$. We
claim that

$\phi^{\prime}(a\otimes f)\geq 0\ .$ (5.8)

To establish this, let $\epsilon>0$. We use Lemma 5.1 to get an element $g\in
G^{\prime}$ such

$\left|a(x)f(t)-\Sigma(g)(x,t)\right|\leq\frac{\epsilon}{2}$ (5.9)

for all $(x,t)\in\Delta\times L$. Using the density of $\theta(H)$ in
${\operatorname{Aff}}(\Delta)$ we choose $h\in H^{+}$ such that
$\epsilon\leq\Sigma(h)(x)<2\epsilon$ for all $x\in\Delta$. Then
$\mathbb{L}(g+[[h]])=0$ and

$\Sigma(g+[[h]])(x,t)\geq\frac{\epsilon}{2}$

for all $(x,t)\in(F\times L)\cup(\Delta\times\\{1\\})$, implying that
$g+[[h]]\in G^{+}$ and hence that
$\phi(g+[[h]])=\phi^{\prime}(\Sigma(g))+\phi([[h]])\geq 0$. It follows from
Observation 5.11 that $\phi([[h]])\leq 2\epsilon$, and we infer that
$\phi^{\prime}(\Sigma(g))\geq-2\epsilon$. Combined with (5.9) it follows that
$\phi^{\prime}(a\otimes f)\geq-3\epsilon$, proving (5.8). In the same way as
in the proof of Lemma 4.8 we deduce from (5.7) and (5.8) that $s\in L$ and
that there is a real number $\lambda(a)$ for each
$a\in{\operatorname{Aff}}(\Delta)$ such that

$\phi^{\prime}(a\otimes f)\ =\ \lambda(a)f(s)$ (5.10)

for all $f\in C_{0}(L)$. The resulting map $a\mapsto\lambda(a)$ is clearly
linear and positive, and $\lambda(1)>0$ since $\phi^{\prime}\neq 0$. Then
$a\mapsto\lambda(1)^{-1}\lambda(a)$ is a state of
${\operatorname{Aff}}(\Delta)$ and there is therefore an $x_{0}\in\Delta$ such
that $\lambda(a)=\lambda(1)a(x_{0})$ for all
$a\in{\operatorname{Aff}}(\Delta)$. Since the elements of $\left\\{a\otimes
f:\ a\in{\operatorname{Aff}}(\Delta),\ f\in C_{0}({L})\right\\}$ span a dense
set in $\mathcal{A}(\Delta\times{L})$ we find that

$\phi^{\prime}(h)\ =\ \lambda(1)h(x_{0},s)\ \ \ \ \forall
h\in\mathcal{A}(\Delta\times{L})\ .$ (5.11)

Note that $\gamma_{*}([[u]])\in G^{\prime}$ and
$\Sigma(\gamma_{*}([[u]]))(x,t)=t^{-1}$. Using (5.11) and the assumptions on
$\phi$ we find that

$\lambda(1)s^{-1}=\phi^{\prime}(\Sigma(\gamma_{*}([[u]])))=\phi(\gamma_{*}([[u]]))=s^{-1}\phi([[u]])=s^{-1}\
,$

implying that $\lambda(1)=1$. Thus

$\phi(g)=\phi^{\prime}(\Sigma(g))=\sum_{m\in\mathbb{Z}}\theta(g_{m})(x_{0})s^{m}$

when $g\in G^{\prime}$. When $g\in G$ is a general element there is an an
$n\in\mathbb{N}\cup\\{0\\}$ such that $g^{\prime}=\gamma_{*}^{n}(g)\in
G^{\prime}$ and then

$\phi(g)=s^{n}\phi(g^{\prime})=s^{n}\sum_{m\in\mathbb{Z}}\theta(g_{m+n})(x_{0})s^{m}=\sum_{m\in\mathbb{Z}}\theta(g_{m})(x_{0})s^{m}\
.$

It remains only to show that $x_{0}\in F$ when $s\neq 1$; which follows from
Lemma 4.3 in the same way as in the proof of Lemma 4.8. ∎

The rest of the proof of Theorem 2.1 for the unbounded case is identical to
the proof from Section 4 for the compact case.

##  6\. Proof of Corollary 2.3 and Corollary 2.4

We will combine Corollary 2.2 with the methods used in Appendix 12 of [Th3]
and Section 6 of [Th4]. Given the UHF algebra $U$ in Corollary 2.3 we write

$U\simeq U_{1}\otimes U_{2}\ ,$

where $U_{1}$ and $U_{2}$ are both (infinite dimensional) UHF-algebras. Given
the collection $\mathbb{I}$ of intervals and simplexes $S_{I},I\in\mathbb{I}$,
in Corollary 2.3 we get from Corollary 5.7 in [Th4] a generalized gauge action
$\alpha^{1}$ on $U_{1}$ with the following properties:

  * •

For each $I\in\mathbb{I}$ and each $\beta\in I\backslash\\{0\\}$ there is a
closed face $F_{I}$ in $S^{\alpha^{1}}_{\beta}$ which is strongly affinely
isomorphic to $S_{I}$.

  * •

For each $\beta\neq 0$ and each $\beta$-KMS state $\omega\in
S^{\alpha^{1}}_{\beta}$ there is a unique norm-convergent decomposition

$\omega=\sum_{I\in\mathbb{I}_{\beta}}\omega_{I}\ $

where $\omega_{I}\in\mathbb{R}^{+}F_{I}$.

That $\alpha^{1}$ is a generalized gauge action means that there is a Bratteli
diagram $\operatorname{Br}$ and a map $F:\operatorname{Br}_{Ar}\to\mathbb{R}$
defined on the set $\operatorname{Br}_{Ar}$ of arrows in $\operatorname{Br}$
which define $\alpha^{1}$ in the following way. Let $\mathcal{P}_{n}$ denote
the set of paths of length $n$ in $\operatorname{Br}$ starting at the top
vertex. Extend the map $F$ to $\mathcal{P}_{n}$ such that

$F(\mu)=\sum_{i=1}^{n}F(a_{i})\ ,$

when $\mu=a_{1}a_{2}\cdots a_{n}$ is made up of the arrows $a_{i}$. The
Bratteli diagram $\operatorname{Br}$ is chosen such that there are finite
dimensional $C^{*}$-subalgebras $\mathbb{F}_{n}$ in $U_{1}$ spanned by a set
of matrix units $E^{n}_{\mu,\mu^{\prime}},\
\mu,\mu^{\prime}\in\mathcal{P}_{n}$, such that
$\mathbb{F}_{n}\subseteq\mathbb{F}_{n+1}$ for all $n$ and

$U_{1}=\overline{\bigcup_{n}\mathbb{F}_{n}}\ .\ $

The flow $\alpha^{1}$ is defined such that

$\alpha^{1}_{t}\left(E^{n}_{\mu,\mu^{\prime}}\right)=e^{i(F(\mu)-F(\mu^{\prime}))t}E^{n}_{\mu,\mu^{\prime}}\
.$

See [Th4]. In order to ensure that $\alpha^{1}$ is $2\pi$-periodic it is
necessary to arrange that $F$ only takes integer values. That this is possible
follows by inspection of the proof in [Th4]; in fact, the only step in the
proof where this is not automatic is in the proof of Lemma 5.3 in [Th4], where
some real numbers $t_{k}$ are chosen. The only crucial property of these
numbers is that they must be sufficiently big and they may therefore be chosen
to be natural numbers. The resulting potential $F$ will then be integer-
valued. Let $\alpha^{2}$ be a flow on $U_{2}$ with the properties specified
for $\alpha$ in Corollary 2.2 and set

$\alpha_{t}=\alpha^{1}_{t}\otimes\alpha^{2}_{t}\ ,$

which we will argue defines a flow on $U_{1}\otimes U_{2}$ with the properties
stated in Corollary 2.3. Since a $\beta$-KMS state for $\alpha$ will restrict
to a $\beta$-KMS state for $\alpha^{2}$ on the tensor factor $U_{2}$, it
follows from Corollary 2.2 that there are no $\beta$-KMS states for $\alpha$
unless $\beta\in K$. Let $\beta\in K$ and let $\omega_{\beta}$ be the unique
$\beta$-KMS state for $\alpha^{2}$. It remains only to show that the map
$\omega\ \mapsto\ \omega\otimes\omega_{\beta}$ is an affine homeomorphism from
$S^{\alpha^{1}}_{\beta}$ onto $S^{\alpha}_{\beta}$. Note that only the
surjectivity is not obvious, and only when $\beta\neq 0$. Consider therefore a
$\beta$-KMS state $\psi\in S^{\alpha}_{\beta}$, $\beta\neq 0$. Let $x,y\in
U_{1},\ b\in U_{2}$ such that $x$ is $\alpha^{1}$-analytic. Then $x\otimes 1$
is $\alpha$-analytic and $\alpha_{i\beta}(x\otimes
1)=\alpha^{1}_{i\beta}(x)\otimes 1$. Hence

$\displaystyle\psi(xy\otimes b)=\psi((x\otimes 1)(y\otimes b))=\psi((y\otimes
b)\alpha_{i\beta}(x\otimes 1))$ $\displaystyle=\psi((y\otimes
b)(\alpha^{1}_{i\beta}(x)\otimes 1))=\psi(y\alpha^{1}_{i\beta}(x)\otimes b)\
.$

Thus, if $b\geq 0$ and $b\neq 0$, the map $U\ni x\mapsto\psi(x\otimes b)$ is a
$\beta$-KMS functional for $\alpha^{1}$, and hence

$\displaystyle\psi(E^{n}_{\mu,\mu^{\prime}}\otimes
b)=\psi(\alpha^{1}_{i\beta}(E^{n}_{\mu,\mu^{\prime}})\otimes
b))=e^{-\beta(F(\mu)-F(\mu^{\prime}))}\psi(E^{n}_{\mu,\mu^{\prime}}\otimes b)\
.$

Since $\beta\neq 0$ it follows that $\psi(E^{n}_{\mu,\mu^{\prime}}\otimes b)\
=\ 0$ when $F(\mu)\neq F(\mu^{\prime})$, and hence that $\psi$ factorises
through the map $Q\otimes\operatorname{id}_{U_{2}}$, where $Q:U_{1}\to
U_{1}^{\alpha^{1}}$ is the conditional expectation onto the fixed point
algebra $U_{1}^{\alpha^{1}}$ of $\alpha^{1}$. Let $a\geq 0$ in $U_{1}$. Then

$U_{2}\ni b\ \mapsto\ \psi(a\otimes b)=\psi(Q(a)\otimes b)$

is a $\beta$-KMS functional for $\alpha^{2}$ since $\psi$ is a $\beta$-KMS
state for $\alpha$ and $Q(a)$ is fixed by $\alpha^{1}$. The uniqueness of
$\omega_{\beta}$ implies therefore that

$\psi(a\otimes b)=\lambda(a)\omega_{\beta}(b)$

for some $\lambda(a)\in\mathbb{R}$ and all $b\in U_{2}$. It is then
straightforward to show that $\lambda(a)$ can be defined for all $a\in U_{1}$,
resulting in a $\beta$-KMS state $\lambda$ for $\alpha^{1}$. This completes
the proof of Corollary 2.3.

Corollary 2.4 follows by choosing the family $\mathbb{I}$ of intervals in
Corollary 2.3 in an appropriate way, e.g. as the collection of all bounded
intervals with rational endpoint, and also the Choquet simplexes $S_{I}$ in an
appropriate way. See Section 6 in [Th4].

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