# Rationally AF algebras and KMS states
of $\mathcal{Z}$-absorbing $\mathrm{C}^{*}$-algebras

George A. Elliott and Yasuhiko Sato
The first named author was supported by a Natural Sciences and Engineering
Research Council of Canada Discovery Grant. The second author was supported by
JSPS (the Grant-in-Aid for Research Activity Start-up 25887031) and the
Department of Mathematical Sciences, Kyushu University.

###### Abstract

In order to realize all possible KMS-bundles on the Jiang-Su algebra, we
introduce a class of $\mathrm{C}^{*}$-algebras which we call rationally
approximately finite dimensional (RAF). Using these, we show that for a given
proper simplex bundle $(S,\pi)$ with a singleton $\pi^{-1}(\\{0\\})$ and a
unital separable monotracial $\mathrm{C}^{*}$-algebra $A$ absorbing the Jiang-
Su algebra tensorially (for instance, the irrational rotation algebra), there
exists a flow on $A$ whose KMS-bundle is isomorphic to $(S,\pi)$.

##  1 Introduction

Approximately finite-dimensional separable $\mathrm{C}^{*}$-algebras (AF
algebras) were classified, roughly fifty years ago, in [35], [17], [8], and
[21], in terms of the Murray-von Neumann semigroup of equivalence classes of
projections—equivalently, the universal enveloping ordered group that has come
to be known (since Murray and von Neumann) as $K_{0}$ (and in the setting of
AF algebras, the dimension group).

A very general class of simple separable $\mathrm{C}^{*}$-algebras, assumed to
be well behaved but satisfying very simple abstract axioms, has now been
classified by a generalization of this invariant; see [47], [48], [49], [62],
[41], [42], [28], [26], [79], [13], [27], [39], and [40], all of which follow
on substantial earlier work over the last thirty to forty years. The axioms
are amenability, absorption of the Jiang-Su algebra $\mathcal{Z}$, and the
Universal Coefficient Theorem (conceivably redundant in the amenable case).
Recall that the tensor product of any $\mathrm{C}^{*}$-algebra with the Jiang-
Su algebra absorbs the Jiang-Su algebra tensorially. In many examples (see
[29]), this property already holds. The invariant, in the stable case, is
$K_{*}=K_{0}\oplus K_{1}$ together with the natural pairing with traces, where
both $K_{0}$ and $K_{1}$ are arbitrary countable abelian groups, and the
traces constitute an arbitrary simplicial cone paired in an arbitrary way with
$K_{0}$ (the natural order in which is determined by this pairing).

At the same time, considerable progress has been made in the last thirty years
in the classification of non-simple $\mathrm{C}^{*}$-algebras beyond the case
of AF algebras. In the infinite ($\mathcal{O}_{\infty}$-stable) case an ideal-
related $KK$-equivalence based isomorphism theorem was outlined in [48]; a
different proof of this was given in [34]. In the finite (non-simple) case,
after earlier results, a $K$-theoretical classification of inductive limits of
finite direct sums of matrix algebras over commutative
$\mathrm{C}^{*}$-algebras (AH, or approximately homogeneous, algebras) with no
dimension growth in the spectra and with the ideal property (every closed two-
sided ideal generated by projections) was given in [37]. The next step in this
direction would be to classify inductive limits of sequences of
$\mathrm{C}^{*}$-subalgebras of matrix algebras over commutative
$\mathrm{C}^{*}$-algebras (ASH, or approximately subhomogeneous, algebras),
with no dimension growth and with the ideal property.

Unexpectedly, in an investigation of KMS-state behavior of one-parameter
automorphism groups of a $\mathrm{C}^{*}$-algebra, along the lines of [5] and
[6], a class of ASH algebras arose which it was possible to classify. This is
the basis of the present paper, which uses this new classification theorem to
construct specified KMS-state phenomena in the Jiang-Su
$\mathrm{C}^{*}$-algebra, and hence in the tensor product of this algebra with
any other monotracial $\mathrm{C}^{*}$-algebra.

The class of algebras in question (see Section 4), to be named rationally AF
algebras (following the terminology of [41] and [42]), or RAF algebras, can be
characterized as those $\mathrm{C}^{*}$-algebras such that the tensor product
with every infinite-dimensional UHF (Glimm uniformly hyperfinite) algebra is
AF.

By [36], every $\mathrm{C}^{*}$-algebra which is both RAF and AH with no
dimension growth is AF. (To see this, one may assume that the algebra is
stable, and then as shown in the proof of Corollary 5.4, below, it has the
ideal property. By [36] it is AT, and since by Lemma 4.2, it has zero $K_{1}$
group, by [76] it is AI. By Corollary 1.3 of [4] any AH algebra with no
dimension growth has real rank zero if it has small eigenvalue variation,
which holds if the tensor product with a UHF algebra has real rank zero. It is
well known that an AI algebra of real rank zero is AF.)

The present class, thus, projects directly into the unknown territory of the
non-simple ASH class.

Interestingly (see Theorem 5.3), the classifying invariant for this class—or,
rather, the Jiang-Su stable members of the class—, at least up to stable
isomorphism, is exactly the same as for the class of AF algebras, namely, the
ordered $K_{0}$-group, which can be an arbitrary countable ordered abelian
group whose tensor product with every dense subgroup of the rational numbers
is a dimension group. (We use the terminology rational dimension group,
following [56] which deals with the simple case. See Definition 3.2, Lemma 3.1
(v), and Theorem 4.4.)

The classification up to isomorphism for non-stable algebras is also almost
the same as for AF algebras, and in the case that there is an approximate unit
consisting of projections (for instance if the algebras are stable or unital),
it is exactly the same (namely, the dimension range). The general case
(appearing already with the non-unital algebras stably isomorphic to the
Jiang-Su algebra) is slightly more subtle. See Corollaries 4.8 and 5.4, which
introduce what we shall call the matrix dimension range.

The proof of Theorem 5.3, the $\mathcal{Z}$-absorbing RAF-algebra
classification theorem, consists of an application of the Winter deformation
technique [82, Proposition 4.5], extended in a simple way beyond the unital
case in which it is at present couched.

Our result concerning KMS-state structure, which follows from the Jiang-Su
absorbing RAF classification (Corollaries 4.8 and 5.4) is Theorem 6.5. (A
precise statement is already given in the Abstract.) The method is similar to
that of [5] and [6], and to that of [32] and [31], and there is some overlap
in the results. In [5] and [6] the $\mathrm{C}^{*}$-algebras on which the
actions are constructed are not precisely identified. In [31], the present
setting but with only the case of a compact bundle, i.e., bounded set of
admissible inverse temperatures, is dealt with—this uses the known simple
$\mathrm{C}^{*}$-algebra classification referred to above. The case of an
infinite Kirchberg algebra is also dealt with in [31], without the compactness
restriction. In [32], the only overlap is the case of a monotracial simple
unital AF algebra.

As a straightforward application of the present paper, we obtain in particular
the following result.

######  Corollary 1.1.

For any irrational rotation algebra, there exists a flow which realizes any
given proper simplex bundle with singleton fibre over $0$ as a KMS-bundle on
it.

##  2 Preliminaries

Let us start with some basic terminology and a fact for ordered abelian groups
with the Riesz interpolation property.

In this paper, a _partially ordered abelian group_ ($G$, $G^{+}$) will mean an
abelian group $G$ with a positive cone $G^{+}$ satisfying $G^{+}+G^{+}\subset
G^{+}$ and $G^{+}\cap-G^{+}=\\{0\\}$. If just the first condition is
satisfied, $(G,G^{+})$ will be called a _pre-ordered abelian group_ (see
[43]). If, in addition to both conditions, $G=G^{+}-G^{+}$, then $(G,G^{+})$
will be called an _ordered abelian group_ (see [69, Definition 5.1.3]). For a
partially ordered (or pre-ordered) abelian group $(G,G^{+})$, we shall denote
by $\leq$ the order relation defined by $g\leq h$ if $h-g\in G^{+}$.

A partially ordered abelian group $(G,G^{+})$ will be said to have the Riesz
(or Birkhoff-Riesz) interpolation property (RIP) if for $g_{0}$, $g_{1}$,
$h_{0}$, $h_{1}\in G$ satisfying $g_{i}\leq h_{j}$ for all $i,j\in\\{0,1\\}$
there exists $a\in G$ such that $g_{i}\leq a\leq h_{j}$ for all
$i,j\in\\{0,1\\}$. (See [7].)

For a supernatural number ${\mathfrak{n}}$ (i.e., a formal infinite product of
finite or infinite powers of prime numbers; see [17], [66]), denote by
$\mathbb{D}_{\mathfrak{n}}$ the set of all rational numbers $p/q$ given by
$p\in\mathbb{Z}$ and $q\in\mathbb{N}$ with $q|{\mathfrak{n}}$, and fix the
positive cone $\mathbb{D}_{\mathfrak{n}}^{+}=\\{d\in\mathbb{D}_{\mathfrak{n}}\
:\ d\geq 0\\}$. Denote by $\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}]$ the group
of Laurent polynomials on $\mathbb{R}$ with coefficients in
$\mathbb{D}_{\mathfrak{n}}$. For a closed subset $F\subset\mathbb{R}$, equip
$\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}]$ with the strict pointwise order,
making it an ordered abelian group:

$\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}]_{F}^{+}=\\{0\\}\cup\\{f\in\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}]\
:\ f(x)>0\text{ for any }x\in F\\}.$

We will have occasion to use the following fundamental lemma in the sequel (in
Lemma 3.5).

######  Lemma 2.1.

Let $F\subset\mathbb{R}$ be a closed subset of $\mathbb{R}$ and
${\mathfrak{n}}$ an infinite supernatural number. Then the ordered abelian
group
$(\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}],\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}]_{F}^{+})$
has the RIP if and only if $F$ is semi-bounded (i.e., either bounded below or
bounded above).

###### Proof.

The “if” part of the statement is a variant of the argument of [78, Section
5]. We include a proof for the reader’s convenience. Suppose that $F$ is semi-
bounded, and set $-F=\\{x\in\mathbb{R}\ :\ -x\in F\\}$. In the case that $F$
is bounded, the Stone-Weierstrass theorem can be applied directly to show the
RIP. So we may assume that $F$ is unbounded. To simplify notation, set
$G=\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}]$ and
$G_{F}^{+}=\mathbb{D}[e^{x},e^{-x}]_{F}^{+}$. It is immediate that
$(G,G_{F}^{+})$ has the RIP if and only if $(G,G_{-F}^{+})$ has. Thus, we may
assume that $F$ is bounded below.

Let $p_{i}$, $q_{j}\in G$, $i,j\in\\{0,1\\}$, be such that $p_{i}\leq q_{j}$
for all $i,j\in\\{0,1\\}$. If $p_{i_{0}}=q_{j_{0}}$ for some
$i_{0},j_{0}\in\\{0,1\\}$, then $a=p_{i_{0}}=q_{j_{0}}$ satisfies $p_{i}\leq
a\leq q_{j}$ for all $i,j\in\\{0,1\\}$. Therefore, we may suppose that
$p_{i}(x)<q_{j}(x)$ for all $i,j\in\\{0,1\\}$ and $x\in F$. As the first step
in the argument which follows, note that for sufficiently large
$N\in\mathbb{N}$ and $R>0$, there exists $a_{0}\in G$ such that

$p_{i}(x)+e^{-Nx}<a_{0}(x)<q_{j}(x)-e^{-Nx},$

for all $i,j\in\\{0,1\\}$ and $x\in F\cap[R,\infty)$.

Let $\mathbb{D}_{\mathfrak{n}}^{\infty}$ denote the infinite direct sum
$\bigoplus_{n\in\mathbb{Z}}\mathbb{D}_{\mathfrak{n}}$ of copies of
$\mathbb{D}_{\mathfrak{n}}$ over $\mathbb{Z}$, and equip
$\mathbb{D}_{\mathfrak{n}}^{\infty}$ with the reverse lexicographic order
$\leq_{\rm lex}$, which means that for $g=(g_{n})_{n\in\mathbb{Z}}$ and
$h=(h_{n})_{n\in\mathbb{Z}}\in\mathbb{D}_{\mathfrak{n}}^{\infty}$, $g\leq_{\rm
lex}h$ if $g_{n}=h_{n}$ for all $n\in\mathbb{Z}$ or $g_{n_{0}}<h_{n_{0}}$ for
$n_{0}=\max\\{n\in\mathbb{Z}\ :\ g_{n}\neq h_{n}\\}$. If $g\leq_{\rm lex}h$
and $g\neq h$, write $g<_{\rm lex}h$. Since $F$ is unbounded (above), it
follows that $(g_{n})_{n\in\mathbb{Z}}<_{\rm lex}(h_{n})_{n\in\mathbb{Z}}$ if,
and only if, there exists $r>0$ such that
$\sum_{n\in\mathbb{Z}}g_{n}e^{nx}<\sum_{n\in\mathbb{Z}}h_{n}e^{nx}$ for any
$x\in F\cap[r,\infty)$. Let $p_{i}^{\infty}=(p_{i,n})_{n\in\mathbb{Z}}$,
$q_{j}^{\infty}=(q_{j,n})_{n\in\mathbb{Z}}\in\mathbb{D}_{\mathfrak{n}}^{\infty}$
denote the coefficient sequences of
$p_{i}(x)=\sum_{n\in\mathbb{Z}}p_{i,n}e^{nx}$ and
$q_{j}(x)=\sum_{n\in\mathbb{Z}}q_{j,n}e^{nx}$, $i,j\in\\{0,1\\}$. Since the
reverse lexicographic order is a total order, reindexing, we may assume that
$p_{0}^{\infty}\leq_{\rm lex}p_{1}^{\infty}<_{\rm lex}q_{0}^{\infty}\leq_{\rm
lex}q_{1}^{\infty}$. Set $n_{0}=\max\\{n\in\mathbb{Z}\ :\ p_{1,n}\neq
q_{0,n}\\}$ and note that $p_{1,n_{0}}<q_{0,n_{0}}$. For $k\in\mathbb{Z}$, set
$\delta_{k}=(\delta_{k,n})_{n\in\mathbb{Z}}\in\mathbb{D}_{\mathfrak{n}}^{\infty}$
where $\delta_{k,n}$ is the Kronecker delta, and set
$a_{0}^{\infty}=p_{1}^{\infty}+\delta_{n_{0}-1}$. Then we have
$p_{1}^{\infty}<_{\rm lex}a_{0}^{\infty}<_{\rm lex}q_{0}^{\infty}$. Let
$N\in\mathbb{N}$ chosen above be chosen sufficiently large that also
$p_{0}^{\infty}+\delta_{-N}\leq_{\rm lex}p_{1}^{\infty}+\delta_{-N}<_{\rm
lex}a_{0}^{\infty}<_{\rm lex}q_{0}^{\infty}-\delta_{-N}\leq
q_{1}^{\infty}-\delta_{-N}$, which is the analogue in
$\mathbb{D}_{\mathfrak{n}}^{\infty}$ of the required condition.

Since by the choice of $a_{0}$,
$e^{Nx}(p_{i}(x)-a_{0}(x))<-1<1<e^{Nx}(q_{j}(x)-a_{0}(x))$ for all
$i,j\in\\{0,1\\}$ and $x\in F\cap[R,\infty)$, there exist $d>0$ and a
continuous function $f:\mathbb{R}\rightarrow\mathbb{R}$ with compact support
such that $e^{Nx}(p_{i}(x)-a_{0}(x))+d<f(x)<e^{Nx}(q_{j}(x)-a_{0}(x))-d$ for
any $x\in F$. By the Stone-Weierstrass theorem (applied to the compact space
$F\cup\\{\infty\\}$ and the subalgebra of Laurent polynomials bounded on this
space), we obtain $a_{1}\in G$ such that $\sup_{x\in F}|f(x)-a_{1}(x)|<d$.
Defining $a\in G$ by $a(x)=e^{-Nx}a_{1}(x)+a_{0}(x)$, we have $p_{i}\leq a\leq
q_{j}$ for all $i,j\in\\{0,1\\}$.

To show the “only if ” part, assume that $F$ is not semi-bounded. Let
$N\in\mathbb{N}$ and $\beta\in F$ be such that $N<e^{\beta}<2N$. Define
$p_{i}$, $q_{j}\in G$, $i,j\in\\{0,1\\}$, by $p_{0}(x)=-1$,
$p_{1}(x)=(N-e^{x})(e^{x}-2N)$, $q_{0}(x)=e^{2x}$, and $q_{1}(x)=N^{2}$ for
$x\in\mathbb{R}$. Then it follows that $p_{i}\leq q_{j}$ for all
$i,j\in\\{0,1\\}$. If there existed $a\in G$ with $p_{i}\leq a\leq q_{j}$ for
all $i,j\in\\{0,1\\}$, then, as $p_{0}\leq a\leq q_{1}$, $a$ would be a
bounded function on $F$. Since $F$ is not semi-bounded, $a$ must be a constant
function, in contradiction with $p_{1}\leq a\leq q_{0}$. ∎

##  3 Rational dimension groups and simplex bundles

The notion of rationally Riesz group was introduced in [56], and it was shown
that a weakly unperforated simple ordered abelian group $G$ is a rationally
Riesz group if and only if $G\otimes\mathbb{D}_{\mathfrak{p}}$ has the Riesz
interpolation property for every infinite supernatural number
${\mathfrak{p}}$. (See [66] for the definition of the supernatural numbers,
which extend the natural numbers, which we shall refer to as finite.) In this
section, we shall consider a modified definition for non-simple ordered
abelian groups (Definition 3.2), and construct an example which gives rise in
a natural way to a given proper simplex bundle (see below), in Proposition
3.6. In the sequel (see Theorem 6.5), proper simplex bundles will be shown to
give rise to KMS-bundles on the Jiang-Su algebra.

Recall that the positive cone $(G\otimes H)^{+}$ of the tensor product of two
partially ordered (or pre-ordered) abelian groups $(G,G^{+})$ and $(H,H^{+})$
is defined as the set of all finite sums of $\\{g\otimes h\ :\ g\in G^{+},h\in
H^{+}\\}$, and note that $(G\otimes H,(G\otimes H)^{+})$ is then a partially
ordered (or pre-ordered) abelian group; see [45]. For convenience, we shall
just write ${G\otimes H}^{+}$. For an abelian group $G$ we shall denote by
$G\otimes 1_{{\bf k}}$ the subgroup of $G\otimes\mathbb{D}_{{\bf k}}$
consisting of $\\{g\otimes 1_{{\bf k}}\ :\ g\in G\\}$, where $1_{{\bf k}}$
denotes the number $1\in\mathbb{D}_{{\bf k}}$.

######  Lemma 3.1.

Let ${\mathfrak{p}}$ and ${\mathfrak{q}}$ be relatively prime supernatural
numbers. For an abelian group $G$, the following fundamental statements hold.

  * (i)

If $G\otimes\mathbb{D}_{\mathfrak{p}}$ and $G\otimes\mathbb{D}_{\mathfrak{q}}$
are torsion-free abelian groups, then so also is $G$.

  * (ii)

If $G$ is torsion free, then the subgroup
$(G\otimes\mathbb{D}_{\mathfrak{p}}\otimes 1_{\mathfrak{q}})\cap(G\otimes
1_{\mathfrak{p}}\otimes\mathbb{D}_{\mathfrak{q}})$ of
$G\otimes\mathbb{D}_{\mathfrak{p}}\otimes\mathbb{D}_{\mathfrak{q}}$ is
isomorphic to $G$ as an abelian group.

  * (iii)

If a partially ordered abelian group $(G,G^{+})$ is such that
$(G\otimes\mathbb{D}_{\mathfrak{p}},{G\otimes\mathbb{D}_{\mathfrak{p}}}^{+})$
and
$(G\otimes\mathbb{D}_{\mathfrak{q}},{G\otimes\mathbb{D}_{\mathfrak{q}}}^{+})$
are torsion-free ordered abelian groups, then $(G,G^{+})$ is an ordered
abelian group (i.e., $G=G^{+}-G^{+}$).

  * (iv)

If $(G,G^{+})$ is an unperforated ordered abelian group (i.e., $ng\geq 0$ for
some $n\in\mathbb{N}$ implies $g\geq 0$; see [2, Definition 6.7.1]), then so
also is
$(G\otimes\mathbb{D}_{\mathfrak{n}},{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
for any supernatural number ${\mathfrak{n}}$.

  * (v)

If $(G,G^{+})$ is an unperforated partially ordered abelian group such that
$(G\otimes\mathbb{D}_{\mathfrak{p}},{G\otimes\mathbb{D}_{\mathfrak{p}}}^{+})$
and
$(G\otimes\mathbb{D}_{\mathfrak{q}},{G\otimes\mathbb{D}_{\mathfrak{q}}}^{+})$
are dimension groups, then $(G,G^{+})$ is an ordered abelian group and
$(G\otimes\mathbb{D}_{\mathfrak{n}},{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
is a dimension group for any infinite supernatural number ${\mathfrak{n}}$.

###### Proof.

(i). Assume that $g\in G$ is a torsion element and $n\in\mathbb{N}$ is the
first number such that $ng=0$. Since $G\otimes\mathbb{D}_{\mathfrak{p}}$ is
torsion free, we have $g\otimes 1_{\mathfrak{p}}=0$ in $\langle
g\rangle\otimes\mathbb{D}_{\mathfrak{p}}$. Since $\langle
g\rangle\otimes\mathbb{D}_{\mathfrak{p}}\cong(\mathbb{Z}/n\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{p}}\cong\mathbb{D}_{\mathfrak{p}}/n\mathbb{D}_{\mathfrak{p}}$,
it follows that $1\in n\mathbb{D}_{\mathfrak{p}}$. Similarly, we have $1\in
n\mathbb{D}_{\mathfrak{q}}$. Because ${\mathfrak{p}}$ and $\mathfrak{q}$ are
relatively prime, this implies that $n=1$ and so $g=ng=0$.

(ii). Since $G$ is torsion free, the canonical embedding $\varphi:G\rightarrow
G\otimes 1_{\mathfrak{p}}\otimes 1_{\mathfrak{q}}$ defined by
$\varphi(g)=g\otimes 1_{\mathfrak{p}}\otimes 1_{\mathfrak{q}}$ for $g\in G$ is
a group isomorphism. Therefore, it suffices to show that
$(G\otimes\mathbb{D}_{\mathfrak{p}}\otimes 1_{\mathfrak{q}})\cap(G\otimes
1_{\mathfrak{p}}\otimes\mathbb{D}_{\mathfrak{q}})=G\otimes
1_{\mathfrak{p}}\otimes 1_{\mathfrak{q}}$. If
$x\in(G\otimes\mathbb{D}_{\mathfrak{p}}\otimes 1_{\mathfrak{q}})\cap(G\otimes
1_{\mathfrak{p}}\otimes\mathbb{D}_{\mathfrak{q}})$, then there exist $k$,
$l\in\mathbb{N}$ such that $1/k\in\mathbb{D}_{\mathfrak{p}}$,
$1/l\in\mathbb{D}_{\mathfrak{q}}$, and $kx,lx\in G\otimes
1_{\mathfrak{p}}\otimes 1_{\mathfrak{q}}$. Since $k$ and $l$ are relatively
prime, there exist $a$, $b\in\mathbb{Z}$ such that $1=ak+bl$. It follows that
$x=akx+blx\in G\otimes 1_{\mathfrak{p}}\otimes 1_{\mathfrak{q}}$. The converse
inclusion $(G\otimes\mathbb{D}_{\mathfrak{p}}\otimes
1_{\mathfrak{q}})\cap(G\otimes
1_{\mathfrak{p}}\otimes\mathbb{D}_{\mathfrak{q}})\supset G\otimes
1_{\mathfrak{p}}\otimes 1_{\mathfrak{q}}$ is trivial.

(iii). Fix $x\in G$. For ${\mathfrak{i}}={\mathfrak{p}}$, ${\mathfrak{q}}$,
since
$(G\otimes\mathbb{D}_{\mathfrak{i}},{G\otimes\mathbb{D}_{\mathfrak{i}}}^{+})$
is an ordered group, there exist $a_{\mathfrak{i}}$,
$b_{\mathfrak{i}}\in{G\otimes\mathbb{D}_{\mathfrak{i}}}^{+}$ such that
$x\otimes 1_{\mathfrak{i}}=a_{\mathfrak{i}}-b_{\mathfrak{i}}$. Let
$n_{\mathfrak{i}}$ be such that
$1/n_{\mathfrak{i}}\in\mathbb{D}_{\mathfrak{i}}$ and
$n_{\mathfrak{i}}a_{\mathfrak{i}}$, $n_{\mathfrak{i}}b_{\mathfrak{i}}\in
G\otimes 1_{\mathfrak{i}}$ for ${\mathfrak{i}}={\mathfrak{p}}$,
${\mathfrak{q}}$. Since $a_{\mathfrak{i}}$,
$b_{\mathfrak{i}}\in{G\otimes\mathbb{D}_{\mathfrak{i}}}^{+}$, there exist
$g_{a_{\mathfrak{i}}}$, $g_{b_{\mathfrak{i}}}\in G^{+}$,
$\mathfrak{i}=\mathfrak{p},\mathfrak{q}$, such that
$n_{\mathfrak{i}}a_{\mathfrak{i}}=g_{a_{\mathfrak{i}}}\otimes
1_{\mathfrak{i}}$ and
$n_{\mathfrak{i}}b_{\mathfrak{i}}=g_{b_{\mathfrak{i}}}\otimes
1_{\mathfrak{i}}$. By (i), $G$ is torsion free, and so from
$n_{\mathfrak{i}}x\otimes
1_{\mathfrak{i}}=(g_{a_{\mathfrak{i}}}-g_{b_{\mathfrak{i}}})\otimes
1_{\mathfrak{i}}$ it follows that
$n_{\mathfrak{i}}x=g_{a_{\mathfrak{i}}}-g_{b_{\mathfrak{i}}}$. Since
$n_{\mathfrak{p}}$ and $n_{\mathfrak{q}}$ are relatively prime, there exist
$c,d\in\mathbb{N}$ such that $1=cn_{\mathfrak{p}}-dn_{\mathfrak{q}}$. Then we
have

$x=(cg_{a_{\mathfrak{p}}}+dg_{b_{\mathfrak{q}}})-(cg_{b_{\mathfrak{p}}}+dg_{a_{\mathfrak{q}}})\in
G^{+}-G^{+}.$

(iv). Since $(G,G^{+})$ and
$(\mathbb{D}_{\mathfrak{n}},{\mathbb{D}_{\mathfrak{n}}}^{+})$ are torsion-free
ordered abelian groups, it is straightforward to check that
$(G\otimes\mathbb{D}_{\mathfrak{n}},{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
is also a torsion-free ordered abelian group. Therefore it suffices to show
that there exists $d\in\mathbb{N}$ such that $1/d\in\mathbb{D}_{\mathfrak{n}}$
and $dx\in{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+}$ if $x\in
G\otimes\mathbb{D}_{\mathfrak{n}}$ and $n\in\mathbb{N}$ satisfy
$nx\in{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+}$. Let $g_{j}\in G$ and
$d_{j}\in\mathbb{D}_{\mathfrak{n}}$, $j=1,2,...,N$, be such that
$\sum_{j=1}^{N}g_{j}\otimes d_{j}=x$. Since
$nx\in{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+}$, there exist $h_{m}\in G_{+}$,
and $e_{m}\in{\mathbb{D}_{\mathfrak{n}}}^{+}$, $m=1,2,...,M$, such that
$nx=\sum_{m=1}^{M}h_{m}\otimes e_{m}$. Choose $d\in\mathbb{N}$ such that
$1/d\in\mathbb{D}_{\mathfrak{n}}$ and $dd_{j}$, $de_{m}\in\mathbb{Z}$ for all
$j=1,2,...,N$ and $m=1,2,...,M$. Then it follows that
$\left(\sum_{j=1}^{N}ndd_{j}g_{j}\right)\otimes
1_{\mathfrak{n}}=\left(\sum_{m=1}^{M}de_{m}h_{m}\right)\otimes
1_{\mathfrak{n}}$. Since $G$ is torsion free, we have
$\sum_{j=1}^{N}ndd_{j}g_{j}=\sum_{m=1}^{M}de_{m}h_{m}\in G^{+}$. Finally,
since $G$ is unperforated, we conclude that
$dx=\left(\sum_{j=1}^{N}dd_{j}g_{j}\right)\otimes
1_{\mathfrak{n}}\in(G\otimes\mathbb{D}_{\mathfrak{n}})^{+}$.

(v). By (i) and (iii), we see that $(G,G^{+})$ is an unperforated ordered
abelian group, and by (iv) so also is
$(G\otimes\mathbb{D}_{\mathfrak{n}},{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$.
Because the Riesz interpolation property in an ordered abelian group is
equivalent to the Riesz decomposition property of [64] (see [7]; see also [16,
Theorem IV 6.2]), it suffices to show that for $x$, $a_{0}$,
$a_{1}\in{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+}$ with $x\leq a_{0}+a_{1}$
there exist two elements $x_{i}\in{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+}$,
$i=0,1$, such that $x=x_{0}+x_{1}$ and $x_{i}\leq a_{i}$ for both $i=0,1$.

Choose $N\in\mathbb{N}$ such that $1/N\in\mathbb{D}_{\mathfrak{n}}$ and $Nx$,
$Na_{i}\in G\otimes 1_{\mathfrak{n}}$, and denote by $g_{x}$, $g_{a_{i}}\in
G^{+}$, $i=0,1$, the elements such that $g_{x}\otimes 1_{\mathfrak{n}}=Nx$ and
$g_{a_{i}}\otimes 1_{\mathfrak{n}}=Na_{i}$ for $i=0,1$. Since $g_{x}\otimes
1_{\mathfrak{n}}\leq(g_{a_{0}}+g_{a_{1}})\otimes 1_{\mathfrak{n}}$, we have
$g_{x}\leq g_{a_{0}}+g_{a_{1}}$, as $G$ is unperforated. By the Riesz
decomposition property of $G\otimes\mathbb{D}_{\mathfrak{p}}$ and
$G\otimes\mathbb{D}_{\mathfrak{q}}$, for
${\mathfrak{j}}={\mathfrak{p}},{\mathfrak{q}}$ there are
$y_{i}^{({\mathfrak{j}})}\in{G\otimes\mathbb{D}_{\mathfrak{j}}}^{+}$, $i=0,1$,
such that $g_{x}\otimes
1_{\mathfrak{j}}=y_{0}^{({\mathfrak{j}})}+y_{1}^{({\mathfrak{j}})}$ and
$y_{i}^{({\mathfrak{j}})}\leq g_{a_{i}}\otimes 1_{\mathfrak{j}}$ for both
$i=0,1$. Choose $N_{\mathfrak{j}}\in\mathbb{N}$,
${\mathfrak{j}}={\mathfrak{p}},{\mathfrak{q}}$, such that
$1/N_{\mathfrak{j}}\in\mathbb{D}_{\mathfrak{j}}$ and
$N_{\mathfrak{j}}y_{i}^{({\mathfrak{j}})}\in G\otimes 1_{\mathfrak{j}}$ for
both $i=0,1$, and denote by $g_{i}^{({\mathfrak{j}})}\in G^{+}$ the element
such that
$N_{\mathfrak{j}}y_{i}^{({\mathfrak{j}})}=g_{i}^{({\mathfrak{j}})}\otimes
1_{\mathfrak{j}}$. Thus we have
$N_{\mathfrak{j}}g_{x}=g_{0}^{({\mathfrak{j}})}+g_{1}^{(\mathfrak{j})}$ for
${\mathfrak{j}}={\mathfrak{p}},{\mathfrak{q}}$. Since $N_{\mathfrak{p}}$ and
$N_{\mathfrak{q}}$ are relatively prime, if ${\mathfrak{n}}$ is infinite,
there exist natural numbers $c$, $d$, and $M$
$(>N_{\mathfrak{p}}N_{\mathfrak{q}})$ such that
$1/(MN)\in\mathbb{D}_{\mathfrak{n}}$ and
$M=cN_{\mathfrak{p}}+dN_{\mathfrak{q}}$. The elements
$x_{i}=(cg_{i}^{({\mathfrak{p}})}+dg_{i}^{({\mathfrak{q}})})\otimes
1/(MN)\in{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+}$, $i=0,1$, then satisfy the
desired conditions. ∎

######  Definition 3.2.

We shall call a partially ordered abelian group $(G,G^{+})$ a _rational
dimension group_ if $(G,G^{+})$ is unperforated and, further,
$(G\otimes\mathbb{D}_{\mathfrak{p}},{G\otimes\mathbb{D}_{\mathfrak{p}}}^{+})$
and
$(G\otimes\mathbb{D}_{\mathfrak{q}},{G\otimes\mathbb{D}_{\mathfrak{q}}}^{+})$
are dimension groups for some pair of relatively prime supernatural numbers
${\mathfrak{p}}$ and ${\mathfrak{q}}$. Note that a partially ordered abelian
group is not necessarily unperforated even if
$G\otimes\mathbb{D}_{\mathfrak{n}}$ is a dimension group for every infinite
supernatural number ${\mathfrak{n}}$. For example, the ordered abelian group
$(\mathbb{Z},S_{n})$ with $S_{n}=\\{0,n,n+1,...\\}$ is not unperforated for
$n\in\mathbb{N}\setminus\\{1\\}$; however,
$(\mathbb{Z}\otimes\mathbb{D}_{\mathfrak{n}},{S_{n}\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
is isomorphic to $(\mathbb{D}_{\mathfrak{n}},{\mathbb{D}_{\mathfrak{n}}}^{+})$
as an ordered abelian group for any infinite supernatural number
${\mathfrak{n}}$.

######  Remark 3.3.

For simple ordered abelian groups, it is enough for the tensor product of an
unperforated ordered group with $\mathbb{D}_{\mathfrak{n}}$ for a single
supernatural number ${\mathfrak{n}}$ to be a dimension group in order for it
to be a rational dimension group (see [56, Proposition 5.7]). However, for
non-simple ordered abelian groups, the following example indicates the
necessity of the two tensor products in the definition above.

Let $p$ and $q$ be two relatively prime numbers, and let ${\mathfrak{u}}$
denote the universal supernatural number, with
$\mathbb{D}_{\mathfrak{u}}=\mathbb{Q}$. Set $H=\mathbb{Q}^{2}$,
$H^{+}=\\{(x,y)\in H\ :\ x\geq 0,y\geq 0\\}$, $a_{0}=(0,0)$, $b_{0}=(1,0)$,
$a_{1}=(1/p,1/p-1)$, and $b_{1}=(1/p,1/p)$ in $H$. Define $(G,G^{+})$ as the
ordered subgroup of $(H,H^{+})$ generated by $\\{a_{1},b_{1}\\}$, which is
unperforated and non-simple. Note that, for any supernatural number
${\mathfrak{n}}$, the ordered abelian group
$(G\otimes\mathbb{D}_{\mathfrak{n}},{G\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
is isomorphic to
$(\mathbb{D}_{\mathfrak{n}}a_{1}+\mathbb{D}_{\mathfrak{n}}b_{1},(\mathbb{D}_{\mathfrak{n}}a_{1}+\mathbb{D}_{\mathfrak{n}}b_{1})\cap
H^{+})$, and that $a_{0}$, $b_{0}\in G$. Then the tensor product
$(G\otimes\mathbb{D}_{\mathfrak{u}},{G\otimes\mathbb{D}_{\mathfrak{u}}}^{+})$
$\cong(H,H^{+})$ is a dimension group. However, if there exists $x\in
G\otimes\mathbb{D}_{q^{\infty}}$ such that $a_{i}\otimes 1_{q^{\infty}}\leq
x\leq b_{j}\otimes 1_{q^{\infty}}$, for $i,j\in\\{0,1\\}$, then $x$ must
correspond to $(1/p,0)$ via the identification of
$G\otimes\mathbb{D}_{q^{\infty}}$ with
$\mathbb{D}_{q^{\infty}}a_{1}+\mathbb{D}_{q^{\infty}}b_{1}$. Since $p$ and $q$
are relatively prime, $x$ cannot be contained in
$G\otimes\mathbb{D}_{q^{\infty}}$. This shows that
$(G\otimes\mathbb{D}_{q^{\infty}},{G\otimes\mathbb{D}_{q^{\infty}}}^{+})$ does
not have the RIP.

Let us recall a few notions concerning simplex bundles, introduced in [5] and
[6] (see also [32] and [31]). A pair $(S,\pi)$ consisting of a second
countable locally compact Hausdorff space $S$ and a continuous map
$\pi:S\rightarrow\mathbb{R}$ is called a _simplex bundle_ if
$\pi^{-1}(\\{\beta\\})$ is compact and has a structure of Choquet simplex for
every $\beta\in\mathbb{R}$. To simplify notation, we shall write
$\pi^{-1}(\beta)$ for $\pi^{-1}(\\{\beta\\})$. Two simplex bundles $(S,\pi)$
and $(S^{\prime},\pi^{\prime})$ are said to be _isomorphic_ if there exists a
homeomorphism $\Phi:S\rightarrow S^{\prime}$ such that
$\pi^{\prime}\circ\Phi=\pi$ and the restriction of $\Phi$ to $\pi^{-1}(\beta)$
is affine for all $\beta\in\mathbb{R}$.

For a simplex bundle $(S,\pi)$, we shall denote by $A(S)$ the set of all
continuous functions $f$ from $S$ to $\mathbb{R}$ such that each restriction
of $f$ to $\pi^{-1}(\beta)$ is affine for any $\beta\in\mathbb{R}$. We also
set

$\displaystyle A_{0}(S)$ $\displaystyle=\\{f\in A(S)\ :\ f\text{ vanishes at
infinity}\\},$ $\displaystyle A_{00}(S)$ $\displaystyle=\\{f\in A(S)\
:\operatorname{supp}(f)\text{ is compact and
}\operatorname{supp}(f)\cap\pi^{-1}(0)=\O\\}.$

A simplex bundle $(S,\pi)$ will be called _proper_ if $\pi$ is a proper map
(i.e. $\pi^{-1}(K)$ is compact for any compact set $K\subset\mathbb{R}$), and
$A(S)$ separates points of $S$.

######  Lemma 3.4.

Let $(S,\pi)$ be a simplex bundle, $S^{\prime}$ a locally compact second
countable Hausdorff space, and $\pi^{\prime}:S^{\prime}\rightarrow\mathbb{R}$
a continuous map. Suppose that $\pi$ is proper and $\Phi$ is a bijective
continuous map from $S$ to $S^{\prime}$ such that $\pi=\pi^{\prime}\circ\Phi$.
Then $\Phi$ is a homeomorphism. In particular, if $(S,\pi)$ is a proper
simplex bundle and the restriction of $\Phi$ to $\pi^{-1}(\beta)$ is affine
for any $\beta\in\mathbb{R}$, then $(S^{\prime},\pi^{\prime})$ is also a
proper simplex bundle.

###### Proof.

For any compact subset $K$ of $S^{\prime}$, since $\pi^{\prime}$ is
continuous, there exists a compact interval $I\subset\mathbb{R}$ such that
$\pi^{\prime}(K)\subset I$, which implies that
$\Phi^{-1}(K)\subset\pi^{-1}(I)$. Since $\pi^{-1}(I)$ is compact, so also is
$\Phi^{-1}(K)$. This shows that $\Phi$ is a proper map. Let $F$ be a closed
subset of $S$. To show $\Phi^{-1}$ is continuous, it suffices to show that
$\Phi(F)$ is closed. For any compact subset $K$ of $S^{\prime}$, it follows
that $\Phi(F)\cap K=\Phi(F\cap\Phi^{-1}(K))$ is compact. Since $S^{\prime}$ is
locally compact and Hausdorff, this shows that $\Phi(F)$ is closed.

As $\pi=\pi^{\prime}\circ\Phi$, it follows that
${\pi^{\prime}}^{-1}(B)=\Phi(\pi^{-1}(B))$ for any subset $B$ of $\mathbb{R}$.
Then $\pi^{\prime}$ is also a proper map. If each restriction of $\Phi$ to
$\pi^{-1}(\beta)$ is affine, then we see that
$A(S^{\prime})=\\{f\circ\Phi^{-1}\ :\ f\in A(S)\\}$ separates points of
$S^{\prime}$. ∎

The concept of a proper simplex bundle was presented as an abstract
characterization of the KMS-bundle of a $\mathrm{C}^{*}$-algebra in [5], [6],
[32], and [31]. On the one hand, it was shown in [5] and [6] (see also [9,
Section 5.3]) that a KMS-bundle for a unital separable
$\mathrm{C}^{*}$-algebra is a proper simplex bundle, and on the other hand, in
[32], it was shown that any proper simplex bundle can be realized as a KMS-
bundle on a given unital simple infinite-dimensional AF-algebra. The
construction in this paper is a variant of [32, Subsection 4.2] for rational
dimension groups (instead of just dimension groups). In the rest of this
section, we shall consider a proper simplex bundle $(S,\pi)$ such that
$\pi^{-1}(0)$ is a singleton $\\{\tau_{S}\\}$.

Let $\mathcal{G}$ be a countable additive subgroup of $\mathbb{R}$ with
$1\in\mathcal{G}$ and let $\mathcal{G}^{\infty}$ denote the infinite direct
sum $\bigoplus_{n\in\mathbb{Z}}\mathcal{G}$ of copies of $\mathcal{G}$ over
$\mathbb{Z}$. Given a sequence $g=(g_{n})_{n\in\mathbb{Z}}$ in
$\mathcal{G}^{\infty}$, define a continuous function $L(g)\in A(S)$ by

$L(g)(s)=\sum_{n\in\mathbb{Z}}g_{n}e^{n\pi(s)}\quad\text{ for }s\in S.$

From now on, we shall denote by $e^{-\pi}$ the function $e^{-\pi(\cdot)}\in
A(S)$, so that $L(g)=\sum_{n\in\mathbb{Z}}g_{n}e^{n\pi}$. Let $\sigma_{S}$
denote the automorphism of $A(S)$ defined by $\sigma_{S}(f)=e^{-\pi}f$ for
$f\in A(S)$, and $\sigma_{\mathcal{G}^{\infty}}$ the automorphism of
$\mathcal{G}^{\infty}$ defined by
$\sigma_{\mathcal{G}^{\infty}}((g_{n})_{n\in\mathbb{Z}})=(g_{n+1})_{n\in\mathbb{Z}}$
for $(g_{n})_{n\in\mathbb{Z}}\in\mathcal{G}^{\infty}$. Note that

$\sigma_{S}\circ L=L\circ\sigma_{\mathcal{G}^{\infty}}.$

For $k\in\mathbb{N}$, choose positive continuous functions $\psi_{k}$,
$\psi_{k+}$, $\psi_{k-}$ from $\mathbb{R}$ into $[0,1]$ such that
$(-\infty,-1/k]\subset\psi_{k-}^{-1}(\\{1\\})$,
$[-1/2k,1/2k]\subset\psi_{k}^{-1}(\\{1\\})$,
$[1/k,\infty)\subset\psi_{k+}^{-1}(\\{1\\})$, and
$\psi_{k-}+\psi_{k}+\psi_{k+}=1_{C(\mathbb{R})}$. Denote by
$\mathcal{G}[e^{-\pi},(1-e^{-\pi})]$ the additive subgroup of $A(S)$ generated
by $\\{ge^{n\pi}(1-e^{-\pi})^{m}\ :\ g\in\mathcal{G},\ m,n\in\mathbb{Z}\\}$.
Consider the countable subgroup of $A(S)$

$\mathcal{G}_{k}^{\infty}=\mathcal{G}[e^{-\pi},(1-e^{-\pi})]\psi_{k-}\circ\pi+L(\mathcal{G}^{\infty})\psi_{k}\circ\pi+\mathcal{G}[e^{-\pi},(1-e^{-\pi})]\psi_{k+}\circ\pi.$

In the same way as in [6, Lemma 2.2] and [32, Property 4.5], choose a
countable subgroup $\mathcal{G}_{00}$ of $A_{00}(S)$ satisfying the following
conditions:

  * (1)

for any $f\in A_{00}(S)$, $\varepsilon>0$, and $N\in\mathbb{N}$ with
$\operatorname{supp}(f)\subset\pi^{-1}((-N,N)\setminus\\{0\\})$, there exists
$g\in\mathcal{G}_{00}$ such that $\sup_{s\in S}|f(x)-g(x)|<\varepsilon$ and
$\operatorname{supp}(g)\subset\pi^{-1}((-N,N)\setminus\\{0\\})$,

  * (2)

$\mathcal{G}_{k}^{\infty}+\mathcal{G}_{00}\subset\mathcal{G}_{k+1}^{\infty}+\mathcal{G}_{00}$
for any $k\in\mathbb{N}$, and

  * (3)

$\sigma_{S}(\mathcal{G}_{00})=(\operatorname{id}_{A(S)}-\sigma_{S})(\mathcal{G}_{00})=\mathcal{G}_{00}$.

Consider the countable subgroup of $A(S)$

$\mathcal{R}(\mathcal{G})=\bigcup_{k=1}^{\infty}\mathcal{G}_{k}^{\infty}+\mathcal{G}_{00}.$

Define positive cones $A(S)^{+}$ and $\mathcal{R}(\mathcal{G})^{+}$ of $A(S)$
and $\mathcal{R}(\mathcal{G})$ by

$\displaystyle A(S)^{+}$ $\displaystyle=\\{0\\}\cup\\{f\in A(S)\ :\
f(x)>0\text{ for any }x\in S\\},$ $\displaystyle\mathcal{R}(\mathcal{G})^{+}$
$\displaystyle=\mathcal{R}(\mathcal{G})\cap A(S)^{+}.$

######  Lemma 3.5.

With $\mathcal{G}$ and $\mathcal{R}(\mathcal{G})$ as above, the following
statements hold.

  * (i)

$(\mathcal{R}(\mathcal{G}),\mathcal{R}(\mathcal{G})^{+})$ is an unperforated
ordered group.

  * (ii)

In the case $\mathcal{G}=\mathbb{Z}$,
$(\mathcal{R}(\mathbb{Z}),\mathcal{R}(\mathbb{Z})^{+})$ is a rational
dimension group.

###### Proof.

(i). It is straightforward to see that
$\mathcal{R}(\mathcal{G})^{+}\cap-\mathcal{R}(\mathcal{G})^{+}=0$ and that the
partially ordered abelian group
$(\mathcal{R}(\mathcal{G}),\mathcal{R}(\mathcal{G})^{+})$ is unperforated. We
must show that
$\mathcal{R}(\mathcal{G})=\mathcal{R}(\mathcal{G})^{+}-\mathcal{R}(\mathcal{G})^{+}$.
For $k\in\mathbb{N}$, let
$g\in\mathcal{G}_{k}^{\infty}+\mathcal{G}_{00}\subset\mathcal{R}(\mathcal{G})$
be given, so that

$g=h_{-}\psi_{k-}\circ\pi+h_{0}\psi_{k}\circ\pi+h_{+}\psi_{k+}\circ\pi+g_{0},$

with $h_{+}$, $h_{-}\in\mathcal{G}[e^{-\pi},(1-e^{-\pi})]$, $h_{0}\in
L(\mathcal{G}^{\infty})$, and $g_{0}\in\mathcal{G}_{00}$. For
$N\in\mathbb{N}$, let us fix the notation $-[N,\infty):=(-\infty,-N]$ and
$+[N,\infty):=[N,\infty)$. With $M$ a large enough natural number, we have
$h_{\pm}(x)\leq e^{\pm M\pi(x)}$ for any $x\in\pi^{-1}(\pm[M,\infty))$. Since
the supports of $h_{0}\psi_{k}\circ\pi$ and $g_{0}$ are compact, and $\pi$ is
proper, there exists $c>0$ such that the function
$f=c(e^{-M\pi}\psi_{k-}\circ\pi+\psi_{k}\circ\pi+e^{M\pi}\psi_{k+}\circ\pi)\in\mathcal{R}(\mathcal{G})^{+}$
satisfies $f-g\in\mathcal{R}(\mathcal{G})^{+}$. Thus,
$g=f-(f-g)\in\mathcal{R}(\mathcal{G})^{+}-\mathcal{R}(\mathcal{G})^{+}$.

(ii). Let ${\mathfrak{n}}$ be an infinite supernatural number and denote by
$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$ the subgroup of $A(S)$
defined by

$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})=\left\\{\sum_{i=1}^{N}d_{i}g_{i}\in
A(S)\ :\ N\in\mathbb{N},\ d_{i}\in\mathbb{D}_{\mathfrak{n}},\
g_{i}\in\mathcal{R}(\mathbb{Z})\right\\}.$

We define a positive cone
$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})^{+}$ by
$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})^{+}=\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})\cap
A(S)^{+}$. First we show that the partially ordered abelian group
$(\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}}),\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})^{+})$
is isomorphic to
$(\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}},{\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
(as a partially ordered abelian group). Note that, by Lemma 3.1 (iv),
$(\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}},{\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
is an unperforated ordered abelian group. Let
$\Phi:\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}}\rightarrow\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$
denote the group homomorphism determined by $\Phi(g\otimes d)=dg$ for
$g\in\mathcal{R}(\mathbb{Z})$ and $d\in\mathbb{D}_{\mathfrak{n}}$, which is
obviously surjective. To show the injectivity of $\Phi$, let $g_{i}$,
$g_{i}^{\prime}\in\mathcal{R}(\mathbb{Z})$ and $d_{i}$,
$d_{i}^{\prime}\in\mathbb{D}_{\mathfrak{n}}$, $i=1,2,...,N$, be such that
$\sum_{i=1}^{N}d_{i}g_{i}=\sum_{i=1}^{N}d_{i}^{\prime}g_{i}^{\prime}$ in
$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$. Choose $d\in\mathbb{N}$
such that $dd_{i}$, $dd_{i}^{\prime}\in\mathbb{Z}$ for all $i=1,2,...,N$ and
$1/d\in\mathbb{D}_{\mathfrak{n}}$. Then it follows that
$\sum_{i=1}^{N}dd_{i}g_{i}=\sum_{i=1}^{N}dd_{i}^{\prime}g_{i}^{\prime}$ in
$\mathcal{R}(\mathbb{Z})$. Thus we have $d(\sum_{i=1}^{N}g_{i}\otimes
d_{i})=d(\sum_{i=1}^{N}g_{i}^{\prime}\otimes d_{i}^{\prime})$ in
$\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}}$. Since
$\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}}$ is torsion free, we
have $\sum_{i=1}^{N}g_{i}\otimes d_{i}=\sum_{i=1}^{N}g_{i}^{\prime}\otimes
d_{i}^{\prime}$. It is trivial to see that
$\Phi({\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}}}^{+})\subset\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})^{+}$.
Since
$(\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}},{\mathcal{R}(\mathbb{Z})\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
is unperforated, we also have the converse inclusion.

What remains to be shown is the RIP of
$(\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}}),\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})^{+})$.
The following argument is essentially same as that in the proof of [32, Lemma
4.6]. Let $f_{i}$,
$g_{j}\in\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$, $i$,
$j\in\\{0,1\\}$, be such that $f_{i}(x)<g_{j}(x)$ for any $x\in S$ and
$i,j\in\\{0,1\\}$. At $\tau_{S}\in\pi^{-1}(0)$, since
$\mathbb{D}_{\mathfrak{n}}$ is dense in $\mathbb{R}$, we obtain
$d\in\mathbb{D}_{\mathfrak{n}}$ such that $f_{i}(\tau_{S})<d<g_{j}(\tau_{S})$
for all $i,j\in\\{0,1\\}$. By the definition of
$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$, there exist
$N\in\mathbb{N}$ and ${\tilde{f}}_{i+}$, ${\tilde{f}}_{i-}$,
${\tilde{g}}_{j+}$,
${\tilde{g}}_{j-}\in\mathbb{D}_{\mathfrak{n}}[e^{-\pi},(1-e^{-\pi})]$,
$i,j\in\\{0,1\\}$ such that ${\tilde{f}}_{i\pm}(x)=f_{i}(x)$ and
${\tilde{g}}_{j\pm}(x)=g_{j}(x)$ for any $i,j\in\\{0,1\\}$ and
$x\in\pi^{-1}(\pm[N,\infty))$. Note that $\pi(\pi^{-1}(\pm[N,\infty))$ are
semi-bounded closed subsets of $\mathbb{R}$, since $\pi$ is a proper map. By
Lemma 2.1, we obtain $e_{-}$,
$e_{+}\in\mathbb{D}_{\mathfrak{n}}[e^{x},e^{-x}]$ such that

$f_{i}(x)<e_{\pm}(\pi(x))<g_{j}(x)\quad\text{ for
}s\in\pi^{-1}(\pm[N,\infty)).$

Set

$\overline{h}(x)=(e_{-}\psi_{k-}+d\psi+e_{+}\psi_{k+})\circ\pi(x)\quad\text{
for }x\in S;$

then it follows that $f_{i}(x)<\overline{h}(x)<g_{j}(x)$ for any
$i,j\in\\{0,1\\}$ and $x\in\pi^{-1}((-\infty,-N]$
$\cup\\{0\\}\cup[N,\infty))$. By [6, Lemma 2.3] and [32, Lemma 4.4], there
exists $\widetilde{h}\in A(S)$ such that $\widetilde{h}(x)=\overline{h}(x)$
for $x\in\pi^{-1}((-\infty,-N]\cup\\{0\\}\cup[N,\infty))$, and
$f_{i}<\widetilde{h}<g_{j}$ on $S$ for all $i,j\in\\{0,1\\}$. Because
$\pi^{-1}([-N,N])$ is compact, there exists $\varepsilon>0$ such that

$f_{i}(x)+\varepsilon<\widetilde{h}(x)<g_{j}(x)-\varepsilon,$

for any $i,j\in\\{0,1\\}$ and $x\in\pi^{-1}([-N,N])$. Set
$\overline{r}(x)=\widetilde{h}(x)-\overline{h}(x)$ for $x\in S$. Then it
follows that
$\operatorname{supp}(\overline{r})\subset\pi^{-1}((-N,N)\setminus\\{0\\})$. By
(1) of the conditions for $\mathcal{G}_{00}$, we have $r\in\mathcal{G}_{00}$
such that $r(x)\approx_{\varepsilon}\overline{r}(x)$ for $x\in S$ and
$\operatorname{supp}(r)\subset\pi^{-1}((-N,N)\setminus\\{0\\})$. The element
$h=\overline{h}+r\in\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$
satisfies $f_{i}<h<g_{j}$ for all $i,j\in\\{0,1\\}$. ∎

For an infinite supernatural number ${\mathfrak{n}}$, consider the ordered
abelian group $(\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}}),$
$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})^{+})$ introduced in the
proof above. Define an ordered group $G_{\mathbb{D}_{\mathfrak{n}}}$ and a
subgroup $G_{\mathbb{Z}}\subset G_{\mathbb{D}_{\mathfrak{n}}}$ by

$\displaystyle G_{\mathbb{D}_{\mathfrak{n}}}$ $\displaystyle=\\{\xi\oplus
g\in(\mathbb{D}_{\mathfrak{n}}^{\infty}\oplus\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}}))\
:\ \exists\varepsilon>0,\text{ for any
}x\in\pi^{-1}((-\varepsilon,\varepsilon)),L(\xi)(x)=g(x)\\},$ $\displaystyle
G_{\mathbb{Z}}$
$\displaystyle=(\mathbb{Z}^{\infty}\oplus\mathcal{R}(\mathbb{Z}))\cap
G_{\mathbb{D}_{\mathfrak{n}}}.$

Define positive cones for $G_{\mathbb{Z}}$ and $G_{\mathbb{D}_{\mathfrak{n}}}$
by

$\displaystyle G_{\mathbb{D}_{\mathfrak{n}}}^{+}$
$\displaystyle=\\{0\\}\cup\\{\xi\oplus g\in G_{\mathbb{D}_{\mathfrak{n}}}\ :\
g\in\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})^{+}\setminus\\{0\\}\\},$
$\displaystyle G_{\mathbb{Z}}^{+}$ $\displaystyle=G_{\mathbb{Z}}\cap
G_{\mathbb{D}_{\mathfrak{n}}}^{+}.$

Note that $(G_{\mathbb{Z}},G_{\mathbb{Z}}^{+})$ is independent of the choice
of infinite supernatural number ${\mathfrak{n}}$. It is straightforward to
check that $(G_{\mathbb{D}_{\mathfrak{n}}},G_{\mathbb{D}_{\mathfrak{n}}}^{+})$
and $(G_{\mathbb{Z}},G_{\mathbb{Z}}^{+})$ are torsion-free ordered abelian
groups such that
$(G_{\mathbb{D}_{\mathfrak{n}}},G_{\mathbb{D}_{\mathfrak{n}}}^{+})$ is
isomorphic to
$(G_{\mathbb{Z}}\otimes\mathbb{D}_{\mathfrak{n}},G_{\mathbb{Z}}\otimes\mathbb{D}_{\mathfrak{n}}^{+})$
(as ordered abelian groups). Since
$\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$ is a dimension group (see
proof of Lemma 3.5), so also is $G_{\mathbb{D}_{\mathfrak{n}}}$, for any
infinite supernatural number ${\mathfrak{n}}$, which means that
$(G_{\mathbb{Z}},G_{\mathbb{Z}}^{+})$ is a rational dimension group.

Define an automorphism $\sigma$ of
$(G_{\mathbb{D}_{\mathfrak{n}}},G_{\mathbb{D}_{\mathfrak{n}}}^{+})$ by

$\sigma(\xi\oplus
g)=(\sigma_{\mathbb{D}_{\mathfrak{n}}^{\infty}}(\xi)\oplus\sigma_{S}(g)),\quad\text{for
}\xi\oplus g\in G_{\mathbb{D}_{\mathfrak{n}}}.$

From $\sigma_{S}(\mathcal{R}(\mathbb{Z})^{+})=\mathcal{R}(\mathbb{Z})^{+}$, it
follows that $\sigma(G_{\mathbb{Z}}^{+})=G_{\mathbb{Z}}^{+}$. Then, using the
same symbol, we may regard $\sigma$ as an automorphism of
$(G_{\mathbb{Z}},G_{\mathbb{Z}}^{+})$.

Set $1_{0}=(\delta_{0,n})_{n\in\mathbb{Z}}\in\mathbb{Z}^{\infty}$, using the
Kronecker delta $\delta_{mn}$ for $m,n\in\mathbb{Z}$, and set $u=(1_{0}\oplus
1)\in G_{\mathbb{Z}}$. Although $u$ is not necessarily an order unit, we may
consider a kind of state space $S_{u}(H)$ for an ordered abelian subgroup
$(H,H^{+})$ of
$(G_{\mathbb{D}_{\mathfrak{n}}},G_{\mathbb{D}_{\mathfrak{n}}}^{+})$ with $u\in
H^{+}$ defined by

$S_{u}(H)=\\{\varphi\ :\ H\rightarrow\mathbb{R}\ :\ \text{ a positive group
homomorphism with }\varphi(u)=1\\}.$

Consider $S_{u}(H)$ with the relative topology from the Cartesian product
$\mathbb{R}^{H}$ (the topology of pointwise convergence). Assuming
$\sigma(H^{+})=H^{+}$, for $\alpha=e^{-\beta}\in\mathbb{R}$, consider the
subset $S_{\sigma}^{\alpha}(H)$ of $S_{u}(H)$ and the bundle $S_{\sigma}(H)$
over $\mathbb{R}$ defined by

$\displaystyle S_{\sigma}^{\alpha}(H)$ $\displaystyle=\\{\varphi\in S_{u}(H)\
:\ \varphi\circ\sigma=\alpha\varphi\\},$ $\displaystyle S_{\sigma}(H)$
$\displaystyle=\\{(\varphi,\beta)\in S_{u}(H)\times\mathbb{R}\ :\ \varphi\in
S_{\sigma}^{\alpha}(H)\\}.$

Although $\mathbb{R}^{H}$ is not locally compact, we shall show that
$S_{\sigma}(G_{\mathbb{Q}})$ is locally compact, in Proposition 3.6 (i) below.
Define a projection $\pi_{\sigma H}:S_{\sigma}(H)\rightarrow\mathbb{R}$ by
$\pi_{\sigma H}(\varphi,\beta)=\beta$. To simplify notation, let us write
$\pi_{\mathbb{Z}}$ and $\pi_{\mathbb{D}_{\mathfrak{n}}}$ for $\pi_{\sigma
G_{\mathbb{Z}}}$ and $\pi_{\sigma G_{\mathbb{D}_{\mathfrak{n}}}}$.

For a supernatural number ${\mathfrak{n}}$, let $\iota_{\mathfrak{n}}$ denote
the canonical embedding of $G_{\mathbb{Z}}$ into
$G_{\mathbb{D}_{\mathfrak{n}}}$, and $\iota_{\mathfrak{n}}^{*}$ the induced
map from $S_{u}(G_{\mathbb{D}_{\mathfrak{n}}})$ to $S_{u}(G_{\mathbb{Z}})$
defined by
$\iota_{\mathfrak{n}}^{*}(\varphi)=\varphi\circ\iota_{\mathfrak{n}}$ for
$\varphi\in S_{u}(G_{\mathbb{D}_{\mathfrak{n}}})$. Define a continuous map
$\iota_{{\mathfrak{n}}\sigma}^{*}$ from
$S_{\sigma}(G_{\mathbb{D}_{\mathfrak{n}}})$ to $S_{\sigma}(G_{\mathbb{Z}})$ by
$\iota_{{\mathfrak{n}}\sigma}^{*}(\varphi,\beta)=(\iota_{\mathfrak{n}}^{*}(\varphi),\beta)$
for $(\varphi,\beta)\in S_{\sigma}(G_{\mathbb{D}_{\mathfrak{n}}})$. For $s\in
S$ and $g\in G_{\mathbb{D}_{\mathfrak{n}}}$ with $g=\xi\oplus f$ for some
$\xi\in\mathbb{D}_{\mathfrak{n}}^{\infty}$ and
$f\in\widetilde{\mathcal{R}}(\mathbb{D}_{\mathfrak{n}})$, set
$\widehat{s}(g)=f(s)$ and regard $\widehat{s}$ as an element of
$S_{\sigma}^{\alpha}(G_{\mathbb{D}_{\mathfrak{n}}})$ for $\alpha=e^{-\pi(s)}$.
Define a continuous map $\Psi_{\mathbb{D}_{\mathfrak{n}}}$ from $S$ to
$S_{\sigma}(G_{\mathbb{D}_{\mathfrak{n}}})$ by
$\Psi_{\mathbb{D}_{\mathfrak{n}}}(s)=(\widehat{s},\pi(s))$ for $s\in S$.

######  Proposition 3.6.

As in Remark 3.3, let ${\mathfrak{u}}$ denote the supernatural number
corresponding to the universal UHF algebra, i.e.,
$\mathbb{D}_{\mathfrak{u}}=\mathbb{Q}$.

  * (i)

The pair $(S_{\sigma}(G_{\mathbb{Q}}),\pi_{\mathbb{Q}})$ is a proper simplex
bundle.

  * (ii)

The induced map
$\iota_{{\mathfrak{u}}\sigma}^{*}:S_{\sigma}(G_{\mathbb{Q}})\rightarrow
S_{\sigma}(G_{\mathbb{Z}})$ is a homeomorphism such that
$\pi_{\mathbb{Z}}\circ\iota_{{\mathfrak{u}}\sigma}^{*}=\pi_{\mathbb{Q}}$ and
the restriction
$\iota_{{\mathfrak{u}}\sigma}^{*}|_{\pi_{\mathbb{Q}}^{-1}(\beta)}$ is affine
for each $\beta\in\mathbb{R}$. Therefore the pair
$(S_{\sigma}(G_{\mathbb{Z}}),\pi_{\mathbb{Z}})$ is also a proper simplex
bundle that is isomorphic to $(S_{\sigma}(G_{\mathbb{Q}}),\pi_{\mathbb{Q}})$.

  * (iii)

$\Psi_{\mathbb{Z}}:S\rightarrow S_{\sigma}(G_{\mathbb{Z}})$ is an isomorphism
of simplex bundles.

To prove the proposition, we shall consider the subgroup of $G_{\mathbb{Q}}$
for which $u$ is an order unit, and its elements of compact support. Let
$\rho:G_{\mathbb{Q}}\rightarrow\widetilde{\mathcal{R}}(\mathbb{Q})$ denote the
standard projection defined by $\rho(\xi\oplus f)=f$. Set
$G_{\mathbb{Q}}^{u+}=\\{g\in G_{\mathbb{Q}}^{+}:\text{ there exists
}N\in\mathbb{N}\text{ such that }x\leq Nu\\}$, $G_{\mathbb{Q}}^{c+}=\\{g\in
G_{\mathbb{Q}}^{+}:\operatorname{supp}(\rho(g))\text{ is compact}\\}$,
$G_{\mathbb{Q}}^{u}=G_{\mathbb{Q}}^{u+}-G_{\mathbb{Q}}^{u+}$, and
$G_{\mathbb{Q}}^{c}=G_{\mathbb{Q}}^{c+}-G_{\mathbb{Q}}^{c+}$. It is
straightforward to check that $(G_{\mathbb{Q}}^{u},G_{\mathbb{Q}}^{u+})$ is an
ordered abelian group with order unit $u$. Since
$(G_{\mathbb{Q}},G_{\mathbb{Q}}^{+})$ is a dimension group, we also see that
$(G_{\mathbb{Q}}^{u},G_{\mathbb{Q}}^{u+})$ has the RIP. Let
$S(G_{\mathbb{Q}}^{u})$ denote the state space of $G_{\mathbb{Q}}^{u}$, a
Choquet simplex (see for example [43, Theorem 10.17]).

To simplify notation, set $Q_{0}=\mathbb{Q}[e^{x},e^{-x}]$ and
$Q=\mathbb{Q}[e^{-x},(1-e^{-x})]$. For
$\xi=(\xi_{n})_{n\in\mathbb{Z}}\in\mathbb{Q}^{\infty}$, define $\Sigma\xi\in
Q_{0}$ by $\Sigma\xi(x)=\sum_{n\in\mathbb{Z}}\xi_{n}e^{nx}$. Let $\varphi\in
S(G_{\mathbb{Q}}^{u})$. Note that, since $\varphi(Ng)=N\varphi(g)$ for any
$g\in G_{\mathbb{Q}}^{u}$ and $N\in\mathbb{N}$, we have $\varphi(\lambda
g)=\lambda\varphi(g)$ for $\lambda\in\mathbb{Q}$ and $g\in
G_{\mathbb{Q}}^{u}$.

######  Lemma 3.7.

Suppose that $\varphi$, $\psi\in S(G_{\mathbb{Q}}^{u})$, $\lambda>0$, and
$\beta\in\mathbb{R}$ satisfy $\psi(g)\leq\lambda\varphi(g)$ for any $g\in
G_{\mathbb{Q}}^{u+}$, and $\varphi(\sigma(g))=e^{-\beta}\varphi(g)$ for any
$g\in G_{\mathbb{Q}}^{c}$. Then it follows that
$\psi(\sigma(g))=e^{-\beta}\psi(g)$ for any $g\in G_{\mathbb{Q}}^{c}$.

###### Proof.

For $\eta=(\eta_{n})_{n\in\mathbb{Z}}$ and
$\xi=(\xi_{n})_{n\in\mathbb{Z}}\in\mathbb{Q}^{\infty}$, let $\eta*\xi$ denote
the convolution product of $\eta$ and $\xi$ (i.e.,
$\eta*\xi=(\sum_{k\in\mathbb{Z}}\eta_{k}\xi_{n-k})_{n}$). For $q\in Q_{0}$ and
$g=(\xi\oplus f)\in G_{\mathbb{Q}}^{c}$, setting
$q^{\infty}=(q_{n})_{n}\in\mathbb{Q}^{\infty}$ such that $\Sigma
q^{\infty}=q$, we define $q\cdot g\in G_{\mathbb{Q}}^{c}$ by $q\cdot
g=(q^{\infty}*\xi\oplus q\circ\pi f)$. Note that, since $\varphi(e^{-x}\cdot
g)=e^{-\beta}\varphi(g)$ for any $g\in G_{\mathbb{Q}}^{c}$, it follows that
$\varphi(q\cdot g)=q(\beta)\varphi(g)$ for any $q\in Q_{0}$ and $g\in
G_{\mathbb{Q}}^{c}$.

Fix $g\in G_{\mathbb{Q}}^{c+}$ and define $\psi_{g}(q)=\psi(q\cdot g)$ for
$q\in Q_{0}$. Thus one obtains the Cauchy-Schwartz inequality

$\psi_{g}(qr)^{2}\leq\psi_{g}(q^{2})\psi_{g}(r^{2})$

for any $q,r\in Q_{0}$. Choose a sequence $\alpha_{n}\in\mathbb{Q}$,
$n\in\mathbb{N}$, such that $\lim\limits_{n\to\infty}\alpha_{n}=e^{-\beta}$,
and set ${\tilde{q}}_{n}=e^{-x}-\alpha_{n}\in Q_{0}$. Then we have

$\displaystyle\psi_{g}({\tilde{q}}_{n})^{2}$
$\displaystyle\leq\psi_{g}({\tilde{q}}_{n}^{2})\psi_{g}(1)\leq\lambda^{-1}\varphi({\tilde{q}}_{n}^{2}\cdot
g)\psi_{g}(1)$
$\displaystyle=\lambda^{-1}{\tilde{q}}_{n}^{2}(\beta)\varphi(g)\psi(g)\rightarrow
0\quad(n\to\infty),$

which implies that $\lim_{n\to\infty}\psi(e^{-x}\cdot g)-\alpha_{n}\psi(g)=0$.
Since $G_{\mathbb{Q}}^{c}=G_{\mathbb{Q}}^{c+}-G_{\mathbb{Q}}^{c+}$, we have
$\psi(e^{-x}\cdot g)=e^{-\beta}\psi(g)$ for any $g\in G_{\mathbb{Q}}^{c}$. ∎

###### Proof of Proposition 3.6.

(i). First we show that
$\pi_{\mathbb{Q}}:S_{\sigma}(G_{\mathbb{Q}})\rightarrow\mathbb{R}$ is a proper
map. Let $K\subset\mathbb{R}$ be a compact set, and
$s_{\lambda}=(\varphi_{\lambda},\beta_{\lambda})\in\pi_{\mathbb{Q}}^{-1}(K)$,
$\lambda\in\Lambda$ be a universal net (see [61] for the definition). For
$g\in G_{\mathbb{Q}}$ such that $\rho(g)$ is bounded on $S$, there exists
$c\in\mathbb{N}$ such that $|\varphi_{\lambda}(g)|<c$ for any
$\lambda\in\Lambda$. Then the net $\varphi_{\lambda}(g)$, $\lambda\in\Lambda$,
converges to a point in $[-c,c]$. For a general $g\in G_{\mathbb{Q}}$, there
are $g_{0}$, $g_{+}$, $g_{-}\in G_{\mathbb{Q}}$ such that
$g=g_{-}+g_{0}+g_{+}$, and $\rho(g_{0})$, $\rho(\sigma^{N}(g_{-}))$, and
$\rho(\sigma^{-N}(g_{+}))$ are bounded on $S$. By
$(\varphi_{\lambda},\beta_{\lambda})\in\pi_{\mathbb{Q}}^{-1}(K)$ and
$\varphi_{\lambda}(g_{\pm})=e^{\pm
N\beta_{\lambda}}\varphi_{\lambda}(\sigma^{\mp N}(g_{\pm}))$, it follows that
$\varphi_{\lambda}(g)$, $\lambda\in\Lambda$, converges in $\mathbb{R}$. Set
$\displaystyle\varphi(g)=\lim_{\lambda\to\infty}\varphi_{\lambda}(g)$. Setting
$\displaystyle\beta=\lim_{\lambda\to\infty}\beta_{\lambda}\in K$, we have
$\varphi(\sigma(g))=e^{-\beta}\varphi(g)$ for any $g\in G_{\mathbb{Q}}$, which
implies that $s_{\lambda}$ converges to
$(\varphi,\beta)\in\pi_{\mathbb{Q}}^{-1}(K)$. Thus, $\pi_{\mathbb{Q}}^{-1}(K)$
is compact.

For $s=(\varphi,\beta)\in S_{\sigma}(G_{\mathbb{Q}})$, we see that
$\pi_{\mathbb{Q}}^{-1}([\beta-\varepsilon,\beta+\varepsilon])$ for
$\varepsilon>0$ is a compact neighbourhood of $S$, which means that
$S_{\sigma}(G_{\mathbb{Q}})$ is locally compact.

It remains to show that $\pi_{\mathbb{Q}}^{-1}(\beta)$ is a Choquet simplex
for any $\beta\in\mathbb{R}$. Fix $\beta\in\mathbb{R}$. Denote by $\iota_{u}$
the canonical embedding of $G_{\mathbb{Q}}^{u}$ into $G_{\mathbb{Q}}$ and
define the induced map $\iota_{u}^{*}:\pi_{\mathbb{Q}}^{-1}(\beta)\rightarrow
S(G_{\mathbb{Q}}^{u})$ by $\iota_{u}^{*}(s)=\varphi\circ\iota_{u}$ for
$s=(\varphi,\beta)\in\pi_{\mathbb{Q}}^{-1}(\beta)$. From the definition, it
follows that $\iota_{u}^{*}$ is affine and continuous in the topology of
pointwise convergence. We see that points of $\pi_{\mathbb{Q}}^{-1}(\beta)$
are separated by $G_{\mathbb{Q}}^{u}$, because any $g\in G_{\mathbb{Q}}$ can
be decomposed as $g=\sigma^{-N}(g_{-})+g_{0}+\sigma^{N}(g_{+})$ for some
$g_{0}$, $g_{-}$, $g_{+}\in G_{\mathbb{Q}}^{u}$ and $N\in\mathbb{N}$. Thus
$\iota_{u}^{*}$ is injective. Since a compact face of a Choquet simplex is
also a Choquet simplex (see [43, Theorem 10.9]), it suffices to show that
$\operatorname{Im}(\iota_{u}^{*})$ is a face of $S(G_{\mathbb{Q}}^{u})$.

Let $\varphi\in\operatorname{Im}(\iota_{u}^{*})$, $\psi$, $\eta\in
S(G_{\mathbb{Q}}^{u})$, and $\lambda\in(0,1)$ satisfy
$\varphi=\lambda\psi+(1-\lambda)\eta$. In the following argument, we first
consider a specific $g\in G_{\mathbb{Q}}^{u}$ with the property
$\rho(g)=\psi_{k+}\circ\pi$, $\psi_{k-}\circ\pi$,
$(\psi_{k+}-\psi_{l+})\circ\pi$, or $(\psi_{k-}-\psi_{l-})\circ\pi$ for $k$,
$l\in\mathbb{N}$ with $k>l$. Set $f_{g}=\rho(g)$, $Q_{g}=\\{q\in Q\ :\
0\oplus(q\circ\pi)f_{g}\in G_{\mathbb{Q}}^{u}\\}$, and
$\psi_{g}(q)=\psi(0\oplus(q\circ\pi)f_{g})$ for $q\in Q_{g}$. Thus we obtain
the Cauchy-Schwartz inequality,
$\psi_{g}(qr)^{2}\leq\psi_{g}(q^{2})\psi_{g}(r^{2})$, for any $q$, $r\in
Q_{g}$ satisfying $qr$, $q^{2}$, $r^{2}\in Q_{g}$. By an argument similar to
that in the proof of Lemma 3.7, we have

$\displaystyle\psi(0\oplus(q\circ\pi)f_{g})=q(\beta)\psi(0\oplus
f_{g})\quad\text{ for any }q\in Q_{g}.$ (1)

For general $g=\xi\oplus f\in G_{\mathbb{Q}}$ represented as
$f=(f_{-}\psi_{k-})\circ\psi+L(\xi)\psi_{k}\circ\pi+(f_{+}\psi_{k+})\circ\pi+f_{0}$
with $f_{+}$, $f_{-}\in Q$, and $f_{0}\in\mathcal{G}_{00}$, define

${\widetilde{\psi}}(g)=f_{-}(\beta)\psi(0\oplus\psi_{k-}\circ\pi)+\psi(\xi\oplus
L(\xi)\psi_{k}\circ\pi)+f_{+}(\beta)\psi(0\oplus\psi_{k+}\circ\pi)+\psi(0\oplus
f_{0}).$

Then ${\widetilde{\psi}}:G_{\mathbb{Q}}\rightarrow\mathbb{R}$ is well defined
(single-valued). Indeed, if two elements $g_{1}$, $g_{2}\in G_{\mathbb{Q}}$
are written as

$g_{i}=\xi^{(i)}\oplus\left((f_{-}^{(i)}\psi_{k_{i}-})\circ\pi+L(\xi^{(i)})(\psi_{k_{i}}\circ\pi)+(f_{+}^{(i)}\psi_{k_{i}+})\circ\pi+f_{0}^{(i)}\right),\quad
i=1,2,$

for some $f_{-}^{(i)}$, $f_{+}^{(i)}\in Q$, and $f_{0}^{(i)}\in G_{00}$, and
if $g_{1}=g_{2}$, then we may assume that $k_{1}\leq k_{2}$ and
$f_{\pm}^{(1)}=f_{\pm}^{(2)}$ without loss of generality. Applying (1) to
$f_{g}=(\psi_{k_{2}\pm}-\psi_{k_{1}\pm})\circ\pi$ we have

$\psi(0\oplus
f_{\pm}(\psi_{k_{2}\pm}-\psi_{k_{1}\pm})\circ\pi)=f_{\pm}(\beta)\psi(0\oplus(\psi_{k_{2}\pm}-\psi_{k_{1}\pm})\circ\pi),$

which implies
${\widetilde{\psi}}(g_{2})-{\widetilde{\psi}}(g_{1})=\psi(g_{2}-g_{1})=0$.
Similarly, applying (1) to $f_{g}=\psi_{k\pm}\circ\pi$ we see that
${\widetilde{\psi}}(g)=\psi(g)$ for any $g\in G_{\mathbb{Q}}^{u}$. It is not
hard to see that ${\widetilde{\psi}}:G_{\mathbb{Q}}\rightarrow\mathbb{R}$ is a
group homomorphism such that ${\widetilde{\psi}}(u)=1$. In the following
paragraph we shall show that ${\widetilde{\psi}}$ is positive.

Let $g=\xi\oplus f\in G_{\mathbb{Q}}$ be such that $f>0$ and let
$k\in\mathbb{N}$, $f_{+}$, $f_{-}\in Q$, and $f_{0}\in\mathcal{G}_{00}$
satisfy
$f=(f_{-}\psi_{k-})\circ\pi+L(\xi)(\psi_{k}\circ\pi)+(f_{+}\psi_{k+}\circ\pi)+f_{0}$.
Let $N\in\mathbb{N}$ and continuous functions $e_{+}$, $e_{-}$ on $\mathbb{R}$
satisfy $\operatorname{supp}(f_{0})\subset[-N,N]$, $0\leq e_{\pm}\leq 1$,
$\operatorname{supp}(e_{\pm})\subset\pm[N,\infty)$, and
$e_{\pm}|_{\pm[N+1,\infty)}=1$. Set

$z=((1-e_{-})f_{-}\psi_{k-})\circ\pi+L(\xi)\psi_{k}\circ\pi+((1-e_{+})f_{+}\psi_{k+})\circ\pi+f_{0}\in
A(S).$

Since $f>0$, we may assume that $N\in\mathbb{N}$, $e_{+}$, and $e_{-}$ also
satisfy $f_{\pm}e_{\pm}\geq 0$, and that $z\geq 0$. Since there exists (a
large) $M\in\mathbb{N}$ such that $e^{\mp Mx}f_{\pm}$ are bounded on
$\pm[0,\infty)$, by approximating $e^{\pm Mx}(1-e_{\pm})$ on $\pm[0,\infty)$
uniformly, we obtain sequences $d_{+n}$, $d_{-n}\in Q$ such that

$\sup_{x\in\pm[0,\infty)}|d_{\pm n}f_{\pm}(x)-(1-e_{\pm})f_{\pm}(x)|<1/n.$

Set

$\displaystyle y_{n}$
$\displaystyle=(f_{-}(1-d_{-n})\psi_{k-})\circ\pi+(f_{+}(1-d_{+n})\psi_{k\pm})\circ\pi\in\widetilde{\mathcal{R}}(\mathbb{Q}),$
$\displaystyle z_{n}$
$\displaystyle=(f_{-}d_{-n}\psi_{k-})\circ\pi+L(\xi)\psi_{k}\circ\pi+(f_{+}d_{+n}\psi_{k+})\circ\pi+f_{0}\in\widetilde{\mathcal{R}}(\mathbb{Q}).$

Then it follows that $0\oplus y_{n}$, $\xi\oplus z_{n}\in G_{\mathbb{Q}}$ and
$g=(0\oplus y_{n})+(\xi\oplus z_{n})$ for any $n\in\mathbb{N}$. Since
$f_{\pm}(1-d_{\pm})$ converge uniformly to $e_{\pm}f_{\pm}$ on
$\pm[0,\infty)$, we have that
$\displaystyle\liminf_{n\to\infty}{\widetilde{\psi}}(0\oplus y_{n})$

$=\liminf_{n\to\infty}f_{-}(1-d_{-n})(\beta)\psi(0\oplus\psi_{k-}\circ\pi)+f_{+}(1-d_{+n})(\beta)\psi(0\oplus\psi_{k+}\circ\pi)\geq
0.$

Since $z_{n}$ converges uniformly to $z\geq 0$ on $S$, and
$\operatorname{supp}(z)$ is compact, we have $\xi\oplus z_{n}\in
G_{\mathbb{Q}}^{u}$ and
$\displaystyle\liminf_{n\to\infty}{\widetilde{\psi}}(\xi\oplus
z_{n})=\liminf_{n\to\infty}\psi(\xi\oplus z_{n})\geq 0$. Thus we conclude that
${\widetilde{\psi}}(g)\geq 0$.

By Lemma 3.7 and $\psi(g)\leq\lambda^{-1}\varphi(g)$ for $g\in
G_{\mathbb{Q}}^{u+}$, we see that
${\widetilde{\psi}}(\sigma(g))=e^{-\beta}{\widetilde{\psi}}(g)$ for any $g\in
G_{\mathbb{Q}}$. From ${\widetilde{\psi}}(g)=\psi(g)$ for any $g\in
G_{\mathbb{Q}}^{u}$, it follows that
$\psi=\iota_{u}^{*}({\widetilde{\psi}})\in\operatorname{Im}(\iota_{u}^{*})$.
By the same argument, we also see that
$\eta\in\operatorname{Im}(\iota_{u}^{*})$, which means that
$\operatorname{Im}(\iota_{u}^{*})$ is a face of $S(G_{\mathbb{Q}}^{u})$, as
asserted, and the proof of (i) is complete.

(ii). For any $g\in G_{\mathbb{Q}}$, consider the smallest number
$N_{g}\in\mathbb{N}$ such that $N_{g}g\in G_{\mathbb{Z}}$. To show injectivity
of $\iota_{{\mathfrak{u}}\sigma}^{*}$, let
$s=(\varphi,\pi_{\mathbb{Q}}(\varphi))$, $t=(\psi,\pi_{\mathbb{Q}}(\psi))\in
S_{\sigma}(G_{\mathbb{Q}})$ satisfy
$\iota_{{\mathfrak{u}}\sigma}^{*}(s)=\iota_{{\mathfrak{u}}\sigma}^{*}(t)$.
Then,

$N_{g}\varphi(g)=\varphi\circ\iota_{\mathfrak{u}}(N_{g}g)=\psi\circ\iota_{\mathfrak{u}}(N_{g}g)=N_{g}\psi(g),$

for any $g\in G_{\mathbb{Q}}$, and it follows that $s=t$. To prove
surjectivity of $\iota_{{\mathfrak{u}}\sigma}^{*}$, let $s=(\varphi,\beta)\in
S_{\sigma}(G_{\mathbb{Z}})$ and define a map
${\tilde{\varphi}}:G_{\mathbb{Q}}\rightarrow\mathbb{R}$ by
${\tilde{\varphi}}(g)=\frac{1}{N_{g}}\varphi(N_{g}g)$ for $g\in
G_{\mathbb{Q}}$. From
${\tilde{\varphi}}(g+h)=\frac{1}{N_{g+h}N_{g}N_{h}}\varphi(N_{g+h}N_{g}N_{h}(g+h))={\tilde{\varphi}}(g)+{\tilde{\varphi}}(h)$
for any $g$, $h\in G_{\mathbb{Q}}$, it follows that ${\tilde{\varphi}}$ is a
group homomorphism. It is straightforward to see that ${\tilde{\varphi}}$ is a
positive group homomorphism satisfying ${\tilde{\varphi}}(u)=1$. By
${\tilde{\varphi}}(\sigma(g))=\frac{1}{N_{g}N_{\sigma(g)}}\varphi(N_{g}N_{\sigma(g)}\sigma(g))=\frac{1}{N_{g}N_{\sigma(g)}}e^{-\beta}\varphi(N_{g}N_{\sigma(g)}g)=e^{-\beta}{\tilde{\varphi}}(g)$
for any $g\in G_{\mathbb{Q}}$, we have $({\tilde{\varphi}},\beta)\in
S_{\sigma}(G_{\mathbb{Q}})$ and
$\iota_{{\mathfrak{u}}\sigma}^{*}(({\tilde{\varphi}},\beta))=(\varphi,\beta)$.

By the same argument as in (i), we see that $S_{\sigma}(G_{\mathbb{Z}})$ is
also locally compact. From the definition of $\iota_{\mathfrak{u}}^{*}$, it
follows that $\iota_{{\mathfrak{u}}\sigma}^{*}$ is a bijective continuous map
such that
$\pi_{\mathbb{Z}}\circ\iota_{{\mathfrak{u}}\sigma}^{*}=\pi_{\mathbb{Q}}$ and
$\iota_{{\mathfrak{u}}\sigma}^{*}|_{\pi_{\mathbb{Q}}^{-1}(\beta)}$ is affine
for any $\beta\in\mathbb{R}$. Because of Lemma 3.4,
$\iota_{{\mathfrak{u}}\sigma}^{*}$ is an isomorphism of simplex bundles and
$(S_{\sigma}(G_{\mathbb{Z}}),\pi_{\mathbb{Z}})$ is also proper.

(iii). As
$\Psi_{\mathbb{Z}}=\iota_{{\mathfrak{u}}\sigma}^{*}\circ\Psi_{\mathbb{Q}}$, it
suffices to show that $\Psi_{\mathbb{Q}}:S\rightarrow
S_{\sigma}(G_{\mathbb{Q}})$ is an isomorphism of simplex bundles. From the
definition, it is straightforward to check that $\Psi_{\mathbb{Q}}$ is
injective and continuous, and the restriction of $\Psi_{\mathbb{Q}}$ to
$\pi^{-1}(\beta)$ is affine for any $\beta\in\mathbb{R}$. By Lemma 3.4, it is
enough to show that $\Psi_{\mathbb{Q}}$ is surjective. However, this is done
in the proof of [32, Lemma 4.12]. We sketch a proof for the convenience of the
reader.

Let $(\varphi,\beta)$ be in $S_{\sigma}(G_{\mathbb{Q}})$. If $g=\xi\oplus f\in
G_{\mathbb{Q}}$ satisfies $f=0\in\widetilde{\mathcal{R}}(\mathbb{Q})$, then it
follows that $ng\leq u$ for any $n\in\mathbb{N}$, which implies
$\varphi(g)=0$. Thus we obtain a positive group homomorphism
$\varphi^{\prime}:\widetilde{\mathcal{R}}(\mathbb{Q})\rightarrow\mathbb{R}$
such that $\varphi^{\prime}\circ\rho=\varphi$. Denote by $A_{\mathbb{R}}(S)$
the real Banach space, with the supremum norm, consisting of functions in
$A(S)$ which have a limit at infinity. Since $\rho(G_{\mathbb{Q}}^{u})\cap
A_{\mathbb{R}}(S)$ is uniformly dense in $A_{\mathbb{R}}(S)$ (see [32, Lemma
4.11]), we can extend $\varphi^{\prime}$ to $A_{\mathbb{R}}(S)$ in such a way
that $|\varphi^{\prime}(f)|\leq\sup_{s\in S}|f(s)|$ for any $f\in
A_{\mathbb{R}}(S)$. By the Hahn-Banach theorem, we can further extend
$\varphi^{\prime}$ to the Banach space $C^{u}(S)=\\{f\in
C(S,\mathbb{R}):\text{ there exists }\lim_{s\to\pm\infty}f(s)\\}$. Since
$\varphi^{\prime}(1)=1$, the extension of $\varphi^{\prime}$ is also a
positive bounded linear functional on $C^{u}(S)$. Thus we obtain a Borel
measure $m_{\varphi}$ on $S$ such that

$\varphi^{\prime}(f)=\int_{S}f(x)\ dm_{\varphi}(x)\quad\text{for any }f\in
C_{0}(S).$

For $y\in\operatorname{Aff}(\pi^{-1}(\beta))$ such that $y>0$, by [32, Lemma
4.4 (2) and Lemma 4.11], we obtain a sequence $g_{yn}=(\xi_{yn}\oplus
f_{yn})\in G_{\mathbb{Q}}$, $n\in\mathbb{N}$, such that
$\operatorname{supp}(f_{yn})$ is compact, $f_{yn}\geq 0$ for any
$n\in\mathbb{N}$, $\sup_{n\in\mathbb{N}}\sup_{s\in S}|f_{yn}(s)|<\infty$, and
$f_{yn}|_{\pi^{-1}(\beta)}$ converges to $y$ uniformly. Let $\omega$ be a free
ultrafilter on $\mathbb{N}$. For $y\in\operatorname{Aff}(\pi^{-1}(\beta))$
with $y>0$, we define
$\displaystyle{\overline{\varphi}}(y)=\lim_{n\to\omega}\varphi^{\prime}(f_{yn})$.
Then ${\overline{\varphi}}$ is independent of the choice of $\xi_{yn}$ and
$f_{yn}$. Indeed, if we suppose that $g_{yn}^{\prime}=(\xi_{yn}^{\prime}\oplus
f_{yn}^{\prime})\in G_{\mathbb{Q}}$ is another choice, then for
$\varepsilon>0$ there exists $N\in\mathbb{N}$ such that
$\sup_{s\in\pi^{-1}(\beta)}|f_{yn}(s)-f_{yn}^{\prime}(s)|<\varepsilon$, for
any $n>N$. Set
$X_{n}=\operatorname{supp}(f_{yn})\cup\operatorname{supp}(f_{yn}^{\prime})$,
which is compact for any $n\in\mathbb{N}$, and let
$q_{m}\in\mathbb{Q}[e^{x},e^{-x}]$, $m\in\mathbb{N}$, be a sequence such that
$q_{m}|_{X_{n}}$ converges to the characteristic function $\chi_{\beta}$ at
$\\{\beta\\}$ and such that $\operatorname{supp}_{s\in X_{n}}|q_{m}(s)|$,
$m\in\mathbb{N}$, is a bounded sequence. Then, by the Lebesgue dominated
convergence theorem,

$\displaystyle\lim_{m\to\infty}\varphi^{\prime}((q_{m}\circ\pi)f_{yn})$
$\displaystyle=\int_{S}(\chi_{\beta}\circ\pi)f_{yn}\
dm_{\varphi}\approx_{\varepsilon}\int_{S}(\chi_{\beta}\circ\pi)f_{yn}^{\prime}\
dm_{\varphi}=\lim_{m\to\infty}\varphi^{\prime}((q_{m}\circ\pi)f_{yn}^{\prime}),$

for any $n\geq N$. It follows that

$|\varphi^{\prime}(f_{yn}-f_{yn}^{\prime})|\leq\limsup_{m\to\infty}\left(\left|\varphi^{\prime}(q_{m}\circ\pi(f_{yn}-f_{yn}^{\prime}))\right|+\left|(1-q_{m}(\beta))\varphi^{\prime}(f_{yn}-f_{yn}^{\prime})\right|\right)<\varepsilon.$

Since $\varepsilon>0$ is arbitrary, we have
$\lim_{n\to\omega}\varphi^{\prime}(f_{yn})=\lim_{n\to\omega}\varphi^{\prime}(f_{yn}^{\prime})$.

Since ${\overline{\varphi}}$ is well defined, we see that
${\overline{\varphi}}(x+y)={\overline{\varphi}}(x)+{\overline{\varphi}}(y)$
for any $x,y\in\operatorname{Aff}(\pi^{-1}(\beta))$ with $x,y>0$. Because
$\operatorname{Aff}(\pi^{-1}(\beta))$ is an ordered abelian group in the
strict ordering, we can extend ${\overline{\varphi}}$ to a positive group
homomorphism from $\operatorname{Aff}(\pi^{-1}(\beta))$ to $\mathbb{R}$. Thus
there exists $s\in\pi^{-1}(\beta)$ and $\lambda\in(0,\infty)$ such that
${\overline{\varphi}}(y)=\lambda y(s)$ for any
$y\in\operatorname{Aff}(\pi^{-1}(\beta))$; see [43, Theorem 7.1], for example.
From the definition of ${\overline{\varphi}}$ it follows that

$\lambda f(s)={\overline{\varphi}}(f|_{\pi^{-1}(\beta)})=\varphi(\xi\oplus
f),$

for any $g=\xi\oplus f\in G_{\mathbb{Q}}$ such that $\operatorname{supp}(f)$
is compact. For $g=\xi\oplus f\in G_{\mathbb{Q}}$ with $f\in A_{0}(S)$ and
$\xi=0$, by the property (1) of $\mathcal{G}_{00}$, there exists a sequence
$f_{n}\in\mathcal{G}_{00}$, $n\in\mathbb{N}$, which converges to $f$
uniformly. Then it follows that $\varphi(0\oplus
f)=\lim_{n\to\infty}\varphi(0\oplus f_{n})=\lambda f(s)$. For $f\in Q$, there
exists a large $N\in\mathbb{N}$ such that $(fe^{\mp Nx}\psi_{k\pm})\circ\pi\in
A_{0}(S)$ for any $k\in\mathbb{N}$. Then we have that
$\varphi(0\oplus(f\psi_{k\pm})\circ\pi)=e^{\pm N\beta}\varphi(0\oplus(e^{\mp
Nx}f\psi_{k\pm})\circ\pi)=\lambda f\psi_{k\pm}(\beta)$, which implies that
$\varphi(g)=\lambda f(s)$ for any $g=\xi\oplus f\in G_{\mathbb{Q}}$. Since
$\varphi(u)=1$, it is clear that $\lambda=1$, and thus we conclude that
$\Psi_{\mathbb{Q}}(s)=(\varphi,\beta)$. ∎

##  4 Rationally AF algebras

In order to realize a given rational dimension group at the level of
$K_{0}$-groups for operator algebras, it seems plausible that a corresponding
$\mathrm{C}^{*}$-algebra should become an AF-algebra after tensoring with a
UHF-algebra. This tensoring procedure is called rationalization in [3], [84].
While the rationalization was determined by a single UHF-algebra in these
previous works, we would like to use the term “rationally” to refer to the
tensor products with two UHF-algebras, that are relatively prime in the
natural sense. (cf. Remark 3.3.)

We prepare some facts and notation concerning $\mathrm{C}^{*}$-algebras. For a
$\mathrm{C}^{*}$-algebra $C$, we let $C^{\sim}$ denote the unitization of $C$,
and $C^{1}$ denote the closed unit ball of $C$. We denote by
$\operatorname{id}_{C}$ the identity automorphism of $C$. For a subset
$F\subset C$, let $P(F)$ denote the set of all projections in $F$ and $F^{+}$
the set of all positive elements in $F$. When $C$ is a unital
$\mathrm{C}^{*}$-algebra, the symbol $1_{C}$ means the unit of $C$. For two
elements $x$, $y\in C$, and $\varepsilon>0$, we use the notation
$x\approx_{\varepsilon}y$ if $\|x-y\|<\varepsilon$. For subsets $F$, $G$ in
$C$, and $\varepsilon>0$, the notation $F\subset_{\varepsilon}G$ means that
for any $x\in F$ there exists $y\in G$ with $y\approx_{\varepsilon}x$.

For a supernatural number ${\mathfrak{n}}$, let $M_{\mathfrak{n}}$ denote the
uniformly hyperfinite (UHF) algebra of type ${\mathfrak{n}}$ (i.e.,
$(K_{0}(M_{\mathfrak{n}}),K_{0}(M_{\mathfrak{n}})^{+},[1_{M_{\mathfrak{n}}}]_{0})\cong(\mathbb{D}_{\mathfrak{n}},\mathbb{D}_{\mathfrak{n}}^{+},1_{\mathbb{D}_{\mathfrak{n}}})$),
and $\mathcal{K}$ the $\mathrm{C}^{*}$-algebra of all compact operators on a
separable Hilbert space. For a natural number $n\in\mathbb{N}$, we let
$\\{e_{ij}^{(n)}\\}_{i,j=1}^{n}$ or $\\{e_{ij}\\}_{i,j=1}^{n}$ denote the set
of canonical matrix units of $M_{n}$.

For two $\mathrm{C}^{*}$-algebras $C$, $D$, and a $*$-homomorphism $\varphi$
from $C$ to $D$, denote the induced map from $K_{0}(C)$ to $K_{0}(D)$, a
homomorphism of pre-ordered abelian groups (the positive cone of $K_{0}$ being
the canonical image of the Murray-von Neumann semigroup—see [69]), by
$\varphi_{*}$.

######  Definition 4.1.

Given a $\mathrm{C}^{*}$-algebra $A$, we shall say that $A$ is _rationally AF_
(RAF) if $A\otimes M_{\mathfrak{p}}$ and $A\otimes M_{\mathfrak{q}}$ are AF-
algebras for some relatively prime pair of supernatural numbers
${\mathfrak{p}}$ and ${\mathfrak{q}}$.

######  Lemma 4.2.

  * (i)

If $A$ is a RAF-algebra, then $A\otimes M_{\mathfrak{n}}$ is approximately
finite dimensional for every infinite supernatural number ${\mathfrak{n}}$.

  * (ii)

Any RAF-algebra has trivial $K_{1}$-group.

###### Proof.

(i). Since $A\otimes M_{n}$ is also a RAF-algebra for any $n\in\mathbb{N}$, to
show $A\otimes M_{\mathfrak{n}}$ is approximately finite dimensional, it
suffices to show that for any finite subset $F$ of $A$ and $\varepsilon>0$
there exists a finite dimensional $\mathrm{C}^{*}$-subalgebra $B$ of $A\otimes
M_{\mathfrak{n}}$ such that $F\otimes
1_{M_{\mathfrak{n}}}\subset_{\varepsilon}B$.

Taking large natural numbers $m_{\mathfrak{p}}$ and $m_{\mathfrak{q}}$ with
$m_{\mathfrak{p}}|{\mathfrak{p}}$ and $m_{\mathfrak{q}}|{\mathfrak{q}}$, we
have finite dimensional $\mathrm{C}^{*}$-subalgebras $A_{\mathfrak{i}}$ of
$A\otimes M_{\mathfrak{i}}$ for ${\mathfrak{i}}={\mathfrak{p}},{\mathfrak{q}}$
such that $F\otimes 1_{M_{\mathfrak{i}}}\subset_{\varepsilon}A_{\mathfrak{i}}$
for both $\mathfrak{i}={\mathfrak{p}},{\mathfrak{q}}$. Since
$m_{\mathfrak{p}}$ and $m_{\mathfrak{q}}$ are relatively prime, there exist
natural numbers $a$, $b$, and $N$ such that
$am_{\mathfrak{p}}+bm_{\mathfrak{q}}=N$ and $N|{\mathfrak{n}}$ (cf. proof of
Lemma 3.1). Therefore there exists a unital embedding $\Phi$ of
$M_{m_{\mathfrak{p}}}\oplus M_{m_{\mathfrak{q}}}$ into $M_{\mathfrak{n}}$.
Identifying $A\otimes(M_{m_{\mathfrak{p}}}\oplus M_{m_{\mathfrak{q}}})$ with
$(A\otimes M_{m_{\mathfrak{p}}})\oplus(A\otimes M_{m_{\mathfrak{q}}})$, we
obtain a finite dimensional $\mathrm{C}^{*}$-subalgebra
$B=(\operatorname{id}_{A}\otimes\Phi)(A_{\mathfrak{p}}\oplus
A_{\mathfrak{q}})$ of $A\otimes M_{\mathfrak{n}}$ which satisfies the above
condition.

(ii). Because of the Künneth theorem [74, Theorem 2.14] (see also [2, Theorem
23.1.3]), we see that $K_{1}(A)\otimes K_{0}(M_{\mathfrak{i}})\cong
K_{1}(A\otimes M_{\mathfrak{i}})=0$ for both
${\mathfrak{i}}={\mathfrak{p}},{\mathfrak{q}}$. From (i) of Lemma 3.1, it
follows that $K_{1}(A)$ is torsion free. Therefore the canonical embedding of
$K_{1}(A)$ into $K_{1}(A)\otimes\mathbb{D}_{\mathfrak{p}}=0$ is injective, and
so $K_{1}(A)=0$. ∎

Note that any RAF-algebra is AF-embeddable, and so it is stably finite. Hence
the ordered $K_{0}$-group of any RAF-algebra is a partially ordered abelian
group. In combination with Lemma 3.1 (iii), the following results show that
the ordered $K_{0}$-group of a $\mathcal{Z}$-absorbing RAF-algebra is an
ordered abelian group.

A typical example of a RAF-algebra is the Jiang-Su algebra $\mathcal{Z}$,
which is constructed as a unital separable simple monotracial RAF-algebra
whose ordered $K_{0}$-group is $(\mathbb{Z},\mathbb{Z}^{+})$. In [38, Theorem
1], Gong, Jiang, and Su showed that the ordered $K_{0}$-group of a unital
simple $\mathcal{Z}$-absorbing $\mathrm{C}^{*}$-algebra is weakly
unperforated. Part (ii) of the following Proposition 4.3 is a variant of their
argument for RAF-algebras in the absence of simplicity.

######  Proposition 4.3.

  * (i)

Let $A$ be a $C^{*}$-algebra and ${\mathfrak{n}}$ a supernatural number. Then
the pre-ordered abelian groups $(K_{0}(A\otimes M_{\mathfrak{n}}),$
$K_{0}(A\otimes M_{\mathfrak{n}})^{+})$ and
$(K_{0}(A)\otimes\mathbb{D}_{\mathfrak{n}},{K_{0}(A)\otimes\mathbb{D}_{\mathfrak{n}}}^{+})$
are isomorphic (in a natural way).

  * (ii)

Let $A$ be a $\mathcal{Z}$-absorbing RAF-algebra. Then the partially ordered
abelian group $(K_{0}(A),K_{0}(A)^{+})$ is unperforated, and, furthermore, is
a rational dimension group.

###### Proof.

(i). It is enough to note that, for any natural number $n$, the Murray-von
Neumann semigroup of $A\otimes M_{n}$ is the same (by definition—with respect
to the map $A\rightarrow A\otimes e_{11}\subset A\otimes M_{n}$) as that of
$A$ (the image of which in $K_{0}(A)$ is the positive cone), and the semigroup
endomorphism corresponding to the map $A\rightarrow A\otimes 1_{M_{n}}\subset
A\otimes M_{n}$ is just multiplication by $n$. (Recall that both the Murray-
von Neumann semigroup functor and the $K_{0}$ functor preserve inductive
limits.)

(ii). If $A$ is a RAF-algebra, then so also is $A\otimes\mathcal{K}$. Since
the ordered $K_{0}$-groups $(K_{0}(A),K_{0}(A)^{+})$ and
$(K_{0}(A\otimes\mathcal{K}),K_{0}(A\otimes\mathcal{K})^{+})$ are isomorphic,
without loss of generality we may assume that $A$ is stable.

Let $\iota$ denote the canonical embedding of $A$ into $A\otimes\mathcal{Z}$
defined by $\iota(a)=a\otimes 1_{\mathcal{Z}}$ for $a\in A$, and $\iota_{*}$
the induced map by $\iota$ from $K_{0}(A)$ to $K_{0}(A\otimes\mathcal{Z})$. As
in the proof of [38, Theorem 1], for $g\in K_{0}(A)$ it is enough to show that
$\iota_{*}(g)\in K_{0}(A\otimes\mathcal{Z})^{+}$ if and only if $ng\in
K_{0}(A)^{+}$ for some $n\in\mathbb{N}$. Indeed, if $x\in
K_{0}(A\otimes\mathcal{Z})$ satisfies $nx\in K_{0}(A\otimes\mathcal{Z})^{+}$
for some $n\in\mathbb{N}$, then, since (by [74]) $\iota_{*}$ is a group
isomorphism, there exists $g_{x}\in K_{0}(A)$ such that $\iota_{*}(g_{x})=x$.
Then we obtain $m\in\mathbb{N}$ such that $mng_{x}\in K_{0}(A)^{+}$ which
implies $x\in K_{0}(A\otimes\mathcal{Z})^{+}$.

Suppose that $g\in K_{0}(A)$ satisfies $\iota_{*}(g)\in
K_{0}(A\otimes\mathcal{Z})^{+}$. Recall that the Jiang-Su algebra is
constructed as the inductive limit $\mathrm{C}^{*}$-algebra
$\lim\limits_{\longrightarrow}(\mathcal{Z}_{p_{n}q_{n}},\varphi_{n})$ of prime
dimension drop algebras $\mathcal{Z}_{p_{n}q_{n}}$, $n\in\mathbb{N}$, and the
connecting maps $\varphi_{n}:\mathcal{Z}_{p_{n}q_{n}}$
$\rightarrow\mathcal{Z}_{p_{n+1}q_{n+1}}$, $n\in\mathbb{N}$, where $p_{n}$ and
$q_{n}$ are relatively prime numbers; see [46]. Note that
$A\otimes\mathcal{Z}_{p_{n}q_{n}}$ is stably finite for each $n\in\mathbb{N}$:
then
$(K_{0}(A\otimes\mathcal{Z}_{p_{n}q_{n}}),K_{0}(A\otimes\mathcal{Z}_{p_{n}q_{n}})^{+})$
is a partially ordered abelian group. By the continuity of the functor $K_{0}$
(see [69, Theorem 6.3.2], for example), there exist embeddings
$\psi_{n}:A\otimes\mathcal{Z}_{p_{n}q_{n}}\rightarrow A\otimes\mathcal{Z}$
such that
$K_{0}(A\otimes\mathcal{Z})^{+}=\bigcup_{n=1}^{\infty}\psi_{n*}(K_{0}(A\otimes\mathcal{Z}_{p_{n}q_{n}})^{+})$.
Then there exist $N\in\mathbb{N}$ and $x\in
K_{0}(A\otimes\mathcal{Z}_{p_{N}q_{N}})^{+}$ such that
$\iota_{*}(g)=\psi_{N*}(x)$. Let $\iota_{N}:A\rightarrow
A\otimes\mathcal{Z}_{p_{N}q_{N}}$ denote the canonical embedding defined by
$\iota_{N}(a)=a\otimes 1_{\mathcal{Z}_{p_{N}q_{N}}}$ for $a\in A$ and
$\operatorname{ev}_{0}:A\otimes\mathcal{Z}_{p_{N}q_{N}}\rightarrow A\otimes
M_{p_{N}}$ the evaluation map determined by $\operatorname{ev}_{0}(f)\otimes
1_{M_{q_{N}}}=f(0)$ for $f\in A\otimes\mathcal{Z}_{p_{N}q_{N}}$. Since
$\operatorname{ev}_{0}\circ\iota_{N}(a)=a\otimes 1_{M_{p_{N}}}$ for any $a\in
A$, it follows that $\operatorname{ev}_{0*}\circ\iota_{N*}(g)=p_{N}g$. Since
$\iota_{N*}(g)=x\in K_{0}(A\otimes\mathcal{Z}_{p_{N}q_{N}})^{+}$, we have
$p_{N}g=\operatorname{ev}_{0*}\circ\iota_{N*}(g)\in K_{0}(A)^{+}$.

Conversely, suppose that $g\in K_{0}(A)$ satisfies $ng\in K_{0}(A)^{+}$ for
some $n\in\mathbb{N}$. Let $p$ and $q\in\mathbb{N}\setminus\\{1\\}$ be
relatively prime natural numbers, and denote by $\iota_{i}:A\rightarrow
A\otimes M_{i^{\infty}}$, $i=p,q$, the canonical embeddings defined by
$\iota_{i}(a)=a\otimes 1_{M_{i^{\infty}}}$ for $a\in A$. Since $K_{0}(A\otimes
M_{i^{\infty}})$, $i=p,q$, are ordered abelian groups, there exist $a_{i}$,
$b_{i}\in K_{0}(A\otimes M_{i^{\infty}})^{+}$ such that
$\iota_{i*}(g)=a_{i}-b_{i}$ for both $i=p,q$. Since $K_{0}(A\otimes
M_{i^{\infty}})$ is unperforated and $n(a_{i}-b_{i})=\iota_{i*}(ng)\in
K_{0}(A\otimes M_{i^{\infty}})^{+}$ for $i=p,q$, it follows that
$a_{i}-b_{i}\in K_{0}(A\otimes M_{i^{\infty}})^{+}$. Because $A\otimes
M_{i^{\infty}}$, $i=p,q$ are stable AF-algebras, we obtain projections
$y_{i}\in A\otimes M_{i^{\infty}}$, $i=p,q$ such that
$[y_{i}]_{0}=a_{i}-b_{i}$. Identifying $A\otimes M_{p^{\infty}}\otimes
M_{q^{\infty}}$ with $A\otimes M_{q^{\infty}}\otimes M_{p^{\infty}}$
canonically, we have $[y_{p}\otimes 1_{M_{q^{\infty}}}]_{0}=[y_{q}\otimes
1_{M_{p^{\infty}}}]_{0}$ in $K_{0}(A\otimes M_{p^{\infty}}\otimes
M_{q^{\infty}})$. Since $A\otimes M_{p^{\infty}}\otimes M_{q^{\infty}}$ is an
AF-algebra, there exists a projection $z\in C([0,1])\otimes A\otimes
M_{p^{\infty}}\otimes M_{q^{\infty}}$ such that $z(0)=y_{p}\otimes
1_{M_{q^{\infty}}}$ and $z(1)=y_{q}\otimes 1_{M_{p^{\infty}}}$. Considering
the $\mathrm{C}^{*}$-algebra

$\mathcal{Z}_{\infty}=\\{f\in C([0,1])\otimes M_{p^{\infty}}\otimes
M_{q^{\infty}}\ :\ f(0)\in M_{p^{\infty}}\otimes 1_{M_{q^{\infty}}},f(1)\in
1_{M_{p^{\infty}}}\otimes M_{q^{\infty}}\\},$

introduced in [70], we can regard $z$ as a projection in
$A\otimes\mathcal{Z}_{\infty}$. Denote by $\iota_{\infty}:A\rightarrow
A\otimes\mathcal{Z}_{\infty}$ the canonical embedding defined by
$\iota_{\infty}(a)=a\otimes 1_{\mathcal{Z}_{\infty}}$ for $a\in A$, and by
$\operatorname{Ev}_{i}:A\otimes\mathcal{Z}_{\infty}\rightarrow A\otimes
M_{i^{\infty}}$, $i=p,q$, the evaluation maps determined by
$\operatorname{Ev}_{p}(f)\otimes 1_{M_{q^{\infty}}}=f(0)$ and
$\operatorname{Ev}_{q}(f)\otimes 1_{M_{p^{\infty}}}=f(1)$ for $f\in
A\otimes\mathcal{Z}_{\infty}$. Since $K_{1}(A\otimes M_{p^{\infty}}\otimes
M_{q^{\infty}})=0$, the induced map
$(\operatorname{Ev}_{p}\oplus\operatorname{Ev}_{q})_{*}:K_{0}(A\otimes\mathcal{Z}_{\infty})\rightarrow
K_{0}(A\otimes M_{p^{\infty}})\oplus K_{0}(A\otimes M_{q^{\infty}})$ is
injective. Since

$(\operatorname{Ev}_{p}\oplus\operatorname{Ev}_{q})_{*}([z]_{0}-\iota_{\infty*}(g))=([y_{p}]_{0}\oplus[y_{q}]_{0})-(\iota_{p*}(g)\oplus\iota_{q*}(g))=0,$

considering the embedding
$\iota_{\mathcal{Z}_{\infty}}:A\otimes\mathcal{Z}_{\infty}\rightarrow
A\otimes\mathcal{Z}$ we conclude that
$\iota_{*}(g)=\iota_{\mathcal{Z}_{\infty}*}\circ\iota_{\infty*}(g)\in
K_{0}(A\otimes\mathcal{Z})^{+}$.

This shows that $(K_{0}(A),K_{0}(A)^{+})$ is an unperforated partially ordered
abelian group. On using once more that $A$ is RAF, the second statement
follows from (i). ∎

Our goal in this section is to provide a construction of RAF-algebras which
exhausts all countable rational dimension groups as $K_{0}$-groups. Here we
emphasize that the non-simple cases of RAF-algebras are required in order to
realize the rational dimension group $(G_{\mathbb{Z}},G_{\mathbb{Z}}^{+})$ of
Section 3. The construction is somewhat analogous to the construction of
simple $\mathrm{C}^{*}$-algebras in [25] and [46].

As a corollary, we also show that a natural addition to the invariant in the
non-stable case is exhausted (Corollary 4.8). In Section 5, we shall show that
the augmented invariant ($K_{0}$ alone in the stable case) is complete.

######  Theorem 4.4.

For any countable rational dimension group $(G,G^{+})$, there exists a
separable stable RAF-algebra $A_{G}$ such that $A_{G}\otimes\mathcal{Z}\cong
A_{G}$ and $(K_{0}(A_{G}),K_{0}(A_{G})^{+})$ is isomorphic to $(G,G^{+})$ as
an ordered abelian group.

Let $(G,G^{+})$ be a countable rational dimension group. For $i=0,1$, we
obtain a stable AF-algebra ${\it A}\raisebox{0.23499pt}{\sf F}_{i}$ such that
$(K_{0}({\it A}\raisebox{0.23499pt}{\sf F}_{i}),K_{0}({\it
A}\raisebox{0.23499pt}{\sf F}_{i})^{+})$ is isomorphic to
$(G\otimes\mathbb{D}_{(2+i)^{\infty}},{G\otimes\mathbb{D}_{(2+i)^{\infty}}}^{+})$
as an ordered abelian group. Because of the Elliott classification theorem
[21] and Proposition 4.3 (i), we can see that

${\it A}\raisebox{0.23499pt}{\sf F}_{0}\otimes M_{3^{\infty}}\cong{\it
A}\raisebox{0.23499pt}{\sf F}_{1}\otimes M_{2^{\infty}}.$

To simplify notation, we suppress the isomorphism between ${\it
A}\raisebox{0.23499pt}{\sf F}_{0}\otimes M_{3^{\infty}}$ and ${\it
A}\raisebox{0.23499pt}{\sf F}_{1}\otimes M_{2^{\infty}}$ in the rest of this
section, and set ${\it A}\raisebox{0.23499pt}{\sf F}={\it
A}\raisebox{0.23499pt}{\sf F}_{0}\otimes M_{3^{\infty}}={\it
A}\raisebox{0.23499pt}{\sf F}_{1}\otimes M_{2^{\infty}}$. We denote by
$\iota_{\infty}^{(i)}$ the canonical embedding of ${\it
A}\raisebox{0.23499pt}{\sf F}_{i}$ into ${\it A}\raisebox{0.23499pt}{\sf F}$
defined by $\iota_{\infty}^{(i)}(a)=a\otimes 1_{M_{(3-i)^{\infty}}}$ for
$i=0,1$ and $a\in{\it A}\raisebox{0.23499pt}{\sf F}_{i}$. The desired RAF-
algebra $A_{G}$ will be constructed as an inductive limit whose building
blocks are type I $\mathrm{C}^{*}$-subalgebras of the following generalized
dimension drop algebra $I_{G}$.

We define a $\mathrm{C}^{*}$-algebra $I_{G}$ by

$I_{G}=\\{f\in C([0,1])\otimes{\it A}\raisebox{0.23499pt}{\sf F}\ :\
f(i)\in\operatorname{Im}(\iota_{\infty}^{(i)})\text{ for both }i=0,1\\}.$

######  Proposition 4.5.

Let $A_{i}$, $i=0,1$, be two AF-embeddable $C^{*}$-algebras. Suppose that
there exist an AF-algebra $A$ and embeddings $\iota^{(i)}$, $i=0,1$, of
$A_{i}$ into $A$. Then the $C^{*}$-algebra $J$ defined by

$J=\\{f\in C([0,1])\otimes A\ :\ f(i)\in\operatorname{Im}(\iota^{(i)})\text{
for both }i=0,1\\}$

has the following properties.

  * (i)

Suppose that the induced maps $\iota_{*}^{(i)}:K_{0}(A_{i})\rightarrow
K_{0}(A)$, $i=0,1$, are injective. Let
$\operatorname{ev}_{i}:J\rightarrow\operatorname{Im}(\iota^{(i)})$, $i=0,1$,
denote the evaluation maps at these two points:
$\operatorname{ev}_{i}(f)=f(i)$. Then, on identifying
$K_{0}(\operatorname{Im}(\iota^{(i)}))$ with
$\operatorname{Im}(\iota_{*}^{(i)})$, the induced maps
$\operatorname{ev}_{i*}:K_{0}(J)\rightarrow\operatorname{Im}(\iota_{*}^{(i)})$,
$i=0,1$, satisfy $\operatorname{ev}_{0*}=\operatorname{ev}_{1*}$ and
$\operatorname{ev}_{0*}:K_{0}(J)\rightarrow\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})$
is a group isomorphism.

  * (ii)

If $\iota^{(i)}$, $i=0,1$, satisfy the assumption of (i) and
$\iota_{*}^{(i)}(K_{0}(A_{i})^{+})=\iota_{*}^{(i)}(K_{0}(A_{i}))\cap
K_{0}(A)^{+}$ for both $i=0,1$, then $\operatorname{ev}_{0*}$ is an
isomorphism of ordered abelian groups from $(K_{0}(J),K_{0}(J)^{+})$ to
$(\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)}),\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})\cap
K_{0}(A)^{+})$.

  * (iii)

In particular, for $A={\it A}\raisebox{0.23499pt}{\sf F}$, $A_{i}={\it
A}\raisebox{0.23499pt}{\sf F}_{i}$, and $\iota^{(i)}=\iota_{\infty}^{(i)}$,
$i=0,1$, it follows that $(K_{0}(I_{G}),K_{0}(I_{G})^{+})$ is isomorphic to
$(G,G^{+})$ as an ordered abelian group.

###### Proof.

(i). Denote by $\iota$ the canonical embedding of $C_{0}((0,1))\otimes A$ into
$J$. From the exact sequence

$0\longrightarrow C_{0}((0,1))\otimes A\xrightarrow{\
\iota\quad}J\xrightarrow{\operatorname{ev}_{0}\oplus\operatorname{ev}_{1}}\operatorname{Im}(\iota^{(0)})\oplus\operatorname{Im}(\iota^{(1)})\longrightarrow
0,$

we obtain the six-term exact sequence of Bott periodicity which implies the
following exact sequence:

$0\longrightarrow
K_{0}(J)\xrightarrow{(\operatorname{ev}_{0}\oplus\operatorname{ev}_{1})_{*}}\operatorname{Im}(\iota_{*}^{(0)})\oplus\operatorname{Im}(\iota_{*}^{(1)})\xrightarrow{\
\partial\quad}K_{0}(A)\longrightarrow 0,$

where $\partial$ is the exponentioal map from
$K_{0}(\operatorname{Im}(\iota^{(0)}))\oplus
K_{0}(\operatorname{Im}(\iota^{(1)}))$ to $K_{1}(C_{0}((0,1))\otimes A)\cong
K_{0}(A)$. Then it is not hard to check that $\partial(x\oplus y)=x-y$ for
$x\in\operatorname{Im}(\iota_{*}^{(0)})$ and
$y\in\operatorname{Im}(\iota_{*}^{(1)})$. Set
$\Phi:\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})\rightarrow\operatorname{Im}(\iota_{*}^{(0)})\oplus\operatorname{Im}(\iota_{*}^{(1)})$
by $\Phi(x)=x\oplus x$ for
$x\in\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})$.
Then the group homomorphism
$(\operatorname{ev}_{0}\oplus\operatorname{ev}_{1})_{*}$ is an isomorphism
onto $\operatorname{Im}(\Phi)$. Thus we see that
$\operatorname{ev}_{0*}=\Phi^{-1}\circ(\operatorname{ev}_{0}\oplus\operatorname{ev}_{1})_{*}=\operatorname{ev}_{1*}$
is a group isomorphism.

(ii). From the assumption
$\iota_{*}^{(i)}(K_{0}(A_{i})^{+})=\iota_{*}^{(i)}(K_{0}(A_{i}))\cap
K_{0}(A)^{+}$, for
$x\in\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})\cap
K_{0}(A)^{+}$, there exist $N\in\mathbb{N}$ and projections $p_{i}\in
A_{i}\otimes M_{N}$, $i=0,1$, such that $\iota_{*}^{(i)}([p_{i}]_{0})=x$ for
both $i=0,1$. Since $A\otimes M_{N}$ is an AF-algebra and
$[\iota^{(0)}\otimes\operatorname{id}_{M_{N}}(p_{0})]_{0}=[\iota^{(1)}\otimes\operatorname{id}_{M_{N}}(p_{1})]_{0}$
in $K_{0}(A)$, there exists a projection ${\widetilde{p}}$ $\in
C([0,1])\otimes A\otimes M_{N}$ such that
${\widetilde{p}}(i)=\iota^{(i)}\otimes\operatorname{id}_{M_{N}}(p_{i})$ for
$i=0,1$. Regarding ${\widetilde{p}}$ as a projection in $J\otimes M_{N}$, we
have that
$(\operatorname{ev}_{0}\oplus\operatorname{ev}_{1})_{*}([{\widetilde{p}}]_{0})=\iota_{*}^{(0)}([p]_{0})\oplus\iota_{*}^{(1)}([p]_{0})=x\oplus
x$. Then it follows that
$x=\Phi^{-1}\circ(\operatorname{ev}_{0}\oplus\operatorname{ev}_{1})_{*}([{\widetilde{p}}]_{0})=\operatorname{ev}_{0*}([{\widetilde{p}}]_{0})\in\operatorname{ev}_{0*}(K_{0}(J)^{+})$,
which implies that
$\operatorname{ev}_{0*}(K_{0}(J)^{+})=\operatorname{Im}(\iota_{*}^{(0)})$
$\cap\operatorname{Im}(\iota_{*}^{(1)})\cap K_{0}(A)^{+}$.

(iii). For $i=0,1$, since $K_{0}(A_{i})$ is torsion free, it follows that the
induced map $\iota_{*}^{(i)}:K_{0}(A_{i})\rightarrow K_{0}(A)\cong
K_{0}(A_{i})\otimes\mathbb{D}_{(3-i)^{\infty}}$ is injective. By Proposition
4.3 (i), it follows that

$\displaystyle(K_{0}(A),K_{0}(A)^{+})$
$\displaystyle\cong(K_{0}(A_{0})\otimes\mathbb{D}_{3^{\infty}},{K_{0}(A_{0})\otimes\mathbb{D}_{3^{\infty}}}^{+})$
$\displaystyle\cong(K_{0}(A_{1})\otimes\mathbb{D}_{2^{\infty}},{K_{0}(A_{1})\otimes\mathbb{D}_{2^{\infty}}}^{+}),$

as ordered abelian groups. Then, for $i=0,1$ and
$x\in\operatorname{Im}(\iota_{*}^{(i)})\cap K_{0}(A)^{+}$, there exists
$d\in\mathbb{N}$ such that $(3-i)^{d}x\in\iota_{*}^{(i)}(K_{0}(A_{i})^{+})$.
Since $(K_{0}(A_{i}),K_{0}(A_{i})^{+})$ is unperforated, it follows that
$x\in\iota_{*}^{(i)}(K_{0}(A_{i})^{+})$, which implies that
$\operatorname{Im}(\iota_{*}^{(i)})\cap
K_{0}(A)^{+}=\iota_{*}^{(i)}(K_{0}(A_{i})^{+})$ for both $i=0,1$. Applying
(ii), we see that $(K_{0}(I_{G}),K_{0}(I_{G})^{+})$ is isomorphic to
$(\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)}),\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})\cap
K_{0}(A)^{+})$.

By Lemma 3.1 (i), any rational dimension group is torsion free, and by Lemma
3.1 (ii), we have
$\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})=G\otimes\mathbb{D}_{2^{\infty}}\otimes
1_{\mathbb{D}_{3^{\infty}}}\cap G\otimes
1_{\mathbb{D}_{2^{\infty}}}\otimes\mathbb{D}_{3^{\infty}}\cong G$. Let
$\Psi:G\rightarrow
G\otimes\mathbb{D}_{2^{\infty}}\otimes\mathbb{D}_{3^{\infty}}$ denote the
positive group homomorphism defined by $\Psi(g)=g\otimes
1_{\mathbb{D}_{2^{\infty}}}\otimes 1_{\mathbb{D}_{3^{\infty}}}$ for $g\in G$.
The group isomorphism of (the proof of) Lemma 3.1 (ii) is just the map $\Psi$.
Furthermore, since $G$ is unperforated it is straightforward to check that
$\Psi(G^{+})=\Psi(G)\cap(G\otimes\mathbb{D}_{2^{\infty}}\otimes\mathbb{D}_{3^{\infty}})^{+}$.
Therefore we conclude that

$\displaystyle(\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)}),\operatorname{Im}(\iota_{*}^{(0)})\cap\operatorname{Im}(\iota_{*}^{(1)})\cap
K_{0}(A)^{+})$
$\displaystyle\cong(\Psi(G),\Psi(G)\cap(G\otimes\mathbb{D}_{2^{\infty}}\otimes\mathbb{D}_{3^{\infty}})^{+})\cong(G,G^{+}).$

∎

Before going into the proof, we collect some necessary well-known facts
concerning finite dimensional $\mathrm{C}^{*}$-algebras.

######  Lemma 4.6.

  * (i)

If natural numbers $P$, $Q$, $R$ and a projection $p\in M_{Q}\otimes M_{R}$
satisfy $R\leq\operatorname{rank}(p)$, $R|\operatorname{rank}(p)$,
$\frac{\operatorname{rank}(p)}{R}<Q$, and
$P|(Q-\frac{\operatorname{rank}(p)}{R})$, then there exists a unitary $u\in
M_{P}\otimes M_{Q}\otimes M_{R}$ such that: for any $\mathrm{C}^{*}$-algebra
$A$, $a\in A$, and $b\in A\otimes M_{P}$,

$\operatorname{Ad}(1_{A^{\sim}}\otimes u)((a\otimes 1_{M_{P}}\otimes
p)\oplus(b\otimes(1_{M_{Q}\otimes M_{R}}-p)))\in A\otimes 1_{M_{P}}\otimes
M_{Q}\otimes 1_{M_{R}}.$

  * (ii)

For $\varepsilon>0$ and $N\in\mathbb{N}$, there exists $\delta>0$ such that:
if $A$ and $B$ are $C^{*}$-subalgebras of a unital $C^{*}$-algebra
$\mathscr{A}$ such that $\dim(B)\leq N$ and $B^{1}\subset_{\delta}A$, then
there exists a unitary $u\in\mathscr{A}$ such that
$u\approx_{\varepsilon}1_{\mathscr{A}}$ and $\operatorname{Ad}u(B)\subset A$.

  * (iii)

For $\varepsilon>0$ and $N\in\mathbb{N}$, there exists $\delta>0$ such that:
if a finite dimensional $C^{*}$-algebra $A$ and two embeddings $\iota_{0}$ and
$\iota_{1}$ of $A$ into a unital $C^{*}$-algebra $\mathscr{A}$ satisfy
$\dim(A)\leq N$ and $\|\iota_{0}(a)-\iota_{1}(a)\|<\delta$ for all $a\in
A^{1}$, then there exists a unitary $u\in\mathscr{A}$ such that
$\operatorname{Ad}u\circ\iota_{0}=\iota_{1}$ and
$\|u-1_{\mathscr{A}}\|<\varepsilon$.

  * (iv)

For $\varepsilon>0$ and $N\in\mathbb{N}$, there exists $\delta>0$ such that:
if $A$, $B$, and $C$ are $C^{*}$-subalgebras of a unital $C^{*}$-algebra
$\mathscr{A}$ satisfying $C\subset A\cap B$, $B^{1}\subset_{\delta}A$, and
$\dim(B),\dim(C)\leq N$, then there exists a unitary $y\in\mathscr{A}$ such
that $y\in C^{\prime}\cap\mathscr{A}$, $\operatorname{Ad}y(B)\subset A$, and
$\|y-1_{\mathscr{A}}\|<\varepsilon$.

  * (v)

Let $A$ and $B$ be $C^{*}$-subalgebras of a unital $C^{*}$-algebra
$\mathscr{A}$. Suppose that $A$ and $B$ are finite dimensional
$C^{*}$-algebras and unitaries $u$, $v\in\mathscr{A}$ satisfy
$\\{u,v\\}\subset_{1/16}B+\mathbb{C}1_{\mathscr{A}}$ and
$\operatorname{Ad}u(A)\cup\operatorname{Ad}v(A)\subset B$. Then there exists a
unitary $w\in B+\mathbb{C}1_{\mathscr{A}}$ such that
$\operatorname{Ad}wu(a)=\operatorname{Ad}v(a)$ for any $a\in A$.

###### Proof.

(i). Set a projection $e\in M_{Q}$ with
$\operatorname{rank}(e)=\frac{\operatorname{rank}(p)}{R}$. By
$P|(Q-\operatorname{rank}(e))$, we obtain a unital embedding
$\iota:M_{P}\rightarrow(1_{M_{Q}}-e)M_{Q}(1_{M_{Q}}-e)$. Since
$P\operatorname{rank}(p)=PR\operatorname{rank}(e)$, there exists a partial
isometry $v\in M_{P}\otimes M_{Q}\otimes M_{R}$ such that
$v^{*}v=1_{M_{P}}\otimes p$ and $vv^{*}=1_{M_{P}}\otimes e\otimes 1_{M_{R}}$.
Denote by $\eta$ the canonical embedding of $M_{P}$ into
$M_{P}\otimes(1_{M_{Q}\otimes M_{R}}-p)$ defined by
$\eta(a)=a\otimes(1_{M_{Q}\otimes M_{R}}-p)$ for $a\in M_{P}$. Since the
multiplicity of $\eta$ is $QR-\operatorname{rank}(p)$, the same as that of
$1_{M_{P}}\otimes\iota\otimes 1_{M_{R}}$, there exists a partial isometry
$w\in M_{P}\otimes M_{Q}\otimes M_{R}$ such that $w^{*}w=\eta(1_{M_{P}})$,
$ww^{*}=1_{M_{P}}\otimes\iota(1_{M_{P}})\otimes 1_{M_{R}}$, and
$\operatorname{Ad}w\circ\eta=1_{M_{P}}\otimes\iota\otimes 1_{M_{R}}$. The
unitary $u=v+w$ in $M_{P}\otimes M_{Q}\otimes M_{R}$ then satisfies the
desired conditions. Indeed, for any $\mathrm{C}^{*}$-algebra $A$, $a$,
$b^{\prime}\in A$, and $x\in M_{P}$,

$\displaystyle\operatorname{Ad}(1_{A^{\sim}}\otimes u)((a\otimes
1_{M_{P}}\otimes p)\oplus(b^{\prime}\otimes x\otimes(1_{M_{Q}\otimes
M_{R}}-p)))$ $\displaystyle=a\otimes 1_{M_{P}}\otimes e\otimes
1_{M_{R}}+b^{\prime}\otimes 1_{M_{P}}\otimes\iota(x)\otimes 1_{M_{R}}\in
A\otimes 1_{M_{P}}\otimes M_{Q}\otimes 1_{M_{R}}.$

(ii). See [16, Lemma III 3.2].

(iii). Since $A$ is finite dimensional, we may identify $A$ with
$\bigoplus_{l=1}^{L}M_{k_{l}}$ for some $k_{1},k_{2},...,k_{L}\in\mathbb{N}$.
For given $\varepsilon>0$ and $N\in\mathbb{N}$ there exists
$\delta\in(0,\varepsilon/4N)$ such that: if two projections $p$ and $q$ in
$\mathscr{A}$ satisfy $\|p-q\|<\delta$ then there exists a partial isometry
$w\in\mathscr{A}$ such that $w^{*}w=p$, $ww^{*}=q$, and
$w\approx_{\varepsilon/4N}p$. Let us show that $\delta$ is as required.
Suppose that $A$, $\iota_{0}$, and $\iota_{1}$ satisfy the hypotheses. Then
there are partial isometries $w_{l}\in\mathscr{A}$, $l=1,2,...,L$, such that
$w_{l}^{*}w_{l}=\iota_{0}(e_{11}^{(l)})$,
$w_{l}w_{l}^{*}=\iota_{1}(e_{11}^{(l)})$, and
$w_{l}\approx_{\varepsilon/4N}\iota_{0}(e_{11}^{(l)})$. Consider the partial
isometry
$w=\sum_{l=1}^{L}\sum_{j=1}^{k_{l}}\iota_{1}(e_{j,1}^{(l)})w_{l}\iota_{0}(e_{1,j}^{(l)})\in\mathscr{A}$.
We have $\operatorname{Ad}w\circ\iota_{0}=\iota_{1}$ and
$w\approx_{\varepsilon/2}\iota_{0}(1_{A})$. Since
$1_{\mathscr{A}}-\iota_{0}(1_{A})\approx_{\delta}1_{\mathscr{A}}-\iota_{1}(1_{A})$,
there exists a partial isometry $v\in\mathscr{A}$ such that
$v^{*}v=1_{\mathscr{A}}-\iota_{0}(1_{A})$,
$vv^{*}=1_{\mathscr{A}}-\iota_{1}(1_{A})$, and
$v\approx_{\varepsilon/2}1_{\mathscr{A}}-\iota_{0}(1_{A})$. The unitary
$u=v+w\in\mathscr{A}$ satisfies the stipulated conditions.

(iv). Applying (iii), we obtain $\delta^{\prime}\in(0,\varepsilon)$ satisfying
the condition of (iii) for $\varepsilon/2>0$ and $N\in\mathbb{N}$. Applying
(ii), we also obtain $\delta>0$ satisfying the condition of (ii) for
$\delta^{\prime}/2>0$ and $N\in\mathbb{N}$. This $\delta$ satisfies the
desired condition. Indeed, if $A$, $B$, and $C$ satisfy the assumption of (iv)
for $\delta>0$ and $N\in\mathbb{N}$, then there exists a unitary
$u\in\mathscr{A}$ such that $u\approx_{\delta^{\prime}/2}1_{\mathscr{A}}$ and
$\operatorname{Ad}u(B)\subset A$, which implies that
$\operatorname{Ad}u(c)\approx_{\delta^{\prime}}c$ for any $c\in C^{1}$. Thus
we obtain a unitary $w\in A+\mathbb{C}1_{\mathscr{A}}$ such that
$\operatorname{Ad}wu(c)=c$ for any $c\in C$ and
$w\approx_{\varepsilon/2}1_{\mathscr{A}}$. The unitary $y=wu\in\mathscr{A}$
satisfies the conditions.

(v). Let $A$ and $\bigoplus_{l=1}^{L}M_{k_{l}}$ be as in the proof of (iii).
Define a unitary $w^{\prime\prime}\in\mathscr{A}$ by
$w^{\prime\prime}=vu^{*}$. By hypothesis, there exists a unitary $w^{\prime}$
in $B+\mathbb{C}1_{\mathscr{A}}$ such that $w^{\prime}\approx_{1/2}w"$. It
follows that
$\operatorname{Ad}w^{\prime}u(e_{11}^{(l)})\approx_{1}\operatorname{Ad}v(e_{11}^{(l)})$
for all $l=1,2,...,L$. Thus there exists a partial isometry $w_{l}\in B$ such
that $w_{l}^{*}w_{l}=\operatorname{Ad}w^{\prime}u(e_{11}^{(l)})$ and
$w_{l}w_{l}^{*}=\operatorname{Ad}v(e_{11}^{(l)})$. Set
${\overline{w}}=\left(\sum_{l=1}^{N}\sum_{j=1}^{k_{l}}\operatorname{Ad}v(e_{j,1}^{(l)})w_{l}\operatorname{Ad}w^{\prime}u(e_{1j}^{(l)})\right)w^{\prime}$;
${\overline{w}}$ is a partial isometry in $B$ such that
$\operatorname{Ad}{\overline{w}}u(a)=\operatorname{Ad}v(a)$ for any $a\in A$.
Since $B$ is finite dimensional, we can extend ${\overline{w}}$ to a unitary
$w$ in $B+\mathbb{C}1_{\mathscr{A}}$ such that
$\operatorname{Ad}wu(a)=\operatorname{Ad}v(a)$ for any $a\in A$. ∎

In the following proof, for a given $\mathrm{C}^{*}$-algebra $A$ and a
Lipschitz continuous function $f\in C([0,1])\otimes A$ we denote by
$\operatorname{Lip}(f)$ the Lipschitz constant of $f$. We shall denote by
$1_{AF}$ and $1_{AF_{i}}$ the units of the unitizations ${\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$ and ${\it A}\raisebox{0.23499pt}{\sf
F}_{i}^{\sim}$.

###### Proof of Theorem 4.4.

Let $F_{n}$, $n\in\mathbb{N}$, be an increasing sequence of finite subsets of
$G^{+}$ whose union is $G^{+}$ with $F_{1}=\\{0\\}$, and $\varepsilon_{n}>0$,
$n\in\mathbb{N}$, be a decreasing sequence such that
$\sum\limits_{n\in\mathbb{N}}\varepsilon_{n}<1$. Let $F_{An}$ (resp.
$F_{{A}_{i}n}$), $n\in\mathbb{N}$, be an increasing sequence of finite subsets
of ${\it A}\raisebox{0.23499pt}{\sf F}^{1}$ (resp. ${\it
A}\raisebox{0.23499pt}{\sf F}_{i}^{1}$) whose union is dense in ${\it
A}\raisebox{0.23499pt}{\sf F}^{1}$ (resp. ${\it A}\raisebox{0.23499pt}{\sf
F}_{i}^{1}$) and $F_{A1}=\\{0\\}$ (resp. $F_{A_{i}1}=\\{0\\}$ for $i=0,1$).
For $n\in\mathbb{N}$, we shall inductively construct numbers $L_{n}$,
$\Lambda_{n}$, ${\widetilde{L}}_{n}\in\mathbb{N}\cup\\{0\\}$, finite
dimensional $\mathrm{C}^{*}$-subalgebras $A_{n}^{(i)},B_{n}^{(i)}\subset{\it
A}\raisebox{0.23499pt}{\sf F}_{i}$, $i=0,1$, and $A_{n}\subset{\it
A}\raisebox{0.23499pt}{\sf F}$, embeddings
$\epsilon_{1}:\\{0\\}\rightarrow{\it A}\raisebox{0.23499pt}{\sf F}$,
$\epsilon_{n}:M_{2^{L_{n}}}\otimes M_{3^{L_{n}}}\rightarrow
A_{n-1}^{\prime}\cap{\it A}\raisebox{0.23499pt}{\sf F}$ (for $n\geq 2$),
$\iota_{n}^{(i)}:A_{n}^{(i)}\rightarrow A_{n}$,
$\kappa_{n}^{(i)}:B_{n}^{(i)}\otimes M_{(3-i)^{\Lambda_{n}}}\rightarrow{\it
A}\raisebox{0.23499pt}{\sf F}$ for $i=0,1$, a $\mathrm{C}^{*}$-subalgebra
$\mathscr{A}_{n}$ of $C([0,1])\otimes A_{n}$, and an injective
$*$-homomorphism $\varphi_{n}:\mathscr{A}_{n-1}\rightarrow\mathscr{A}_{n}$
(where $\mathscr{A}_{0}=\\{0\\}$) satisfying the following conditions:

  * (1, $n$)

$\Lambda_{n}>L_{n}+\Lambda_{n-1}$,
${\widetilde{L}}_{n}>L_{n}\geq\Lambda_{n-1}$,

  * (2, $n$)

for $i=0,1$, on regarding $(K_{0}({\it A}\raisebox{0.23499pt}{\sf
F}_{i}),K_{0}({\it A}\raisebox{0.23499pt}{\sf F}_{i})^{+})$ as
$(G\otimes\mathbb{D}_{(2+i)^{\infty}},{G\otimes\mathbb{D}_{(2+i)^{\infty}}}^{+})$,
one has $F_{n}\otimes 1_{(2+i)^{\infty}}\subset K_{0}(A_{n}^{(i)})^{+}$,
$A_{n-1}^{(i)}\subset A_{n}^{(i)}\subset B_{n}^{(i)}$, and
$F_{A_{i}n}\subset_{\varepsilon_{n}}A_{n}^{(i)}$,

  * (3, $n$)

if $n=2n^{\prime}+i_{n}$ for some $n^{\prime}\in\mathbb{N}\cup\\{0\\}$ and
$i_{n}\in\\{0,1\\}$, then there exists a unitary $U_{n}$ in ${\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$ such that

$U_{n}\approx_{\varepsilon_{n}}1_{AF}\text{ and
}(A_{n-1}+\mathbb{C}1_{AF})\operatorname{Im}(\epsilon_{n})\subset\operatorname{Ad}U_{n}(A_{n}^{(1-i_{n})}\otimes
M_{(2+i_{n})^{{\widetilde{L}}_{n}}})=A_{n},$

  * (4, $n$)

$\epsilon_{n}(1_{(M_{2^{L_{n}}}\otimes M_{3^{L_{n}}})})a=a$ for any $a\in
A_{n-1}$, and $F_{An}\subset_{\varepsilon_{n}}A_{n}^{1}$,

  * (5, $n$)

$\mathscr{A}_{n}$ is defined by

$\mathscr{A}_{n}=\\{f\in C([0,1])\otimes A_{n}\ :\
f(i)\in\operatorname{Im}(\iota_{n}^{(i)})\text{ for }i=0,1\\},$

  * (6, $n$)

there are unitaries $V_{n}^{(i)}$, $W_{n}^{(i)}\in{\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$, $i=0,1$, such that

$\iota_{n}^{(i)}=\operatorname{Ad}V_{n}^{(i)}\circ\iota_{\infty}^{(i)}|_{A_{n}^{(i)}},\quad\kappa_{n}^{(i)}(b\otimes
1_{M_{(3-i)^{\Lambda_{n}}}})=\operatorname{Ad}W_{n}^{(i)}(b\otimes
1_{M_{(3-i)^{\infty}}}),$

for any $i=0,1$ and $b\in B_{n}^{(i)}$, and

$V_{n}^{(i)}\in_{\varepsilon_{n}}A_{n}+\mathbb{C}1_{AF}\quad\text{ and }\quad
W_{n}^{(i)}\in_{\frac{\varepsilon_{n}}{4}}\operatorname{Im}(\kappa_{n}^{(i)})+\mathbb{C}1_{AF},$

  * (7, $n$)

$A_{n}\subset\operatorname{Im}(\kappa_{n}^{(i)})$ for $i=0,1$, and

$\iota_{n}^{(i)}(a)=\kappa_{n}^{(i)}(a\otimes
1_{M_{(3-i)^{\Lambda_{n}}}})\quad\text{for any }i=0,1,\text{ and }a\in
A_{n}^{(i)},$

  * (8, $n$)

one has

$\operatorname{Lip}(\varphi_{n}(f))\leq\operatorname{Lip}(f)/2+2\varepsilon_{n}$

for any Lipschitz continuous function $f\in\mathscr{A}_{n-1}$ with $\|f\|\leq
1$.

For $n=1$ and $i=0,1$, setting $\Lambda_{0}=0$,
$A_{0}^{(i)}=\\{0\\}\subset{\it A}\raisebox{0.23499pt}{\sf F}_{i}$, and
$A_{0}=\\{0\\}\subset{\it A}\raisebox{0.23499pt}{\sf F}$, define $L_{1}=0$,
$\Lambda_{1}={\widetilde{L}}_{1}=1$, $A_{1}=\\{0\\}\subset{\it
A}\raisebox{0.23499pt}{\sf F}$, $A_{1}^{(i)}=B_{1}^{(i)}=\\{0\\}\subset{\it
A}\raisebox{0.23499pt}{\sf F}_{i}$, $U_{1}=V_{1}^{(i)}=W_{1}^{(i)}=1_{AF}$,
and $\mathscr{A}_{0}=\mathscr{A}_{1}=\\{0\\}$. These choices satisfy (1,
1)–(8, 1).

For the induction, assume that $L_{n}$, $\Lambda_{n}$, ${\widetilde{L}}_{n}$,
$A^{(i)}_{n}$, $B_{n}^{(i)}$, $A_{n}$, $\epsilon_{n}$, $\iota_{n}^{(i)}$,
$\kappa_{n}^{(i)}$, $\mathscr{A}_{n}$, and $\varphi_{n}$ satisfying (1,
$n$)–(8, $n$) have been constructed. Choose $L_{n+1}\in\mathbb{N}$ large
enough that $2^{L_{n+1}}>3^{\Lambda_{n}}$. Then there are $k_{n+1}^{\prime}$,
$l_{n+1}^{\prime}\in\mathbb{N}$ such that $k_{n+1}^{\prime}<3^{\Lambda_{n}}$,
$l_{n+1}^{\prime}<2^{\Lambda_{n}}$,
$3^{\Lambda_{n}}|2^{L_{n+1}}-k_{n+1}^{\prime}$, and
$2^{\Lambda_{n}}|3^{L_{n+1}}-l_{n+1}^{\prime}$. Set
$N_{n+1}=L_{n+1}+\Lambda_{n}$, $k_{n+1}=3^{L_{n+1}}k_{n+1}^{\prime}$,
$l_{n+1}=2^{L_{n+1}}l_{n+1}^{\prime}$, and $m_{n+1}=6^{L_{n+1}}$. Note that
from $k_{n+1}^{\prime}<3^{\Lambda_{n}}\leq 2^{L_{n+1}}-k_{n+1}^{\prime}$ and
$l_{n+1}^{\prime}<2^{\Lambda_{n}}\leq 3^{L_{n+1}}-l_{n+1}^{\prime}$ it follows
that

$m_{n+1}-(k_{n+1}+l_{n+1})=(3^{L_{n+1}}-l_{n+1}^{\prime})(2^{L_{n+1}}-k_{n+1}^{\prime})-l_{n+1}^{\prime}k_{n+1}^{\prime}>0.$

Set
$N=(\dim(B_{n}^{(0)})+\dim(B_{n}^{(1)})+\dim(A_{n}))6^{2L_{n+1}}\in\mathbb{N}$,
and $\delta_{n+1}^{(0)}=\varepsilon_{n+1}/8$. Applying Lemma 4.6 (ii), (iii),
(iv) inductively to $\delta_{n+1}^{(k)}$ $(=\varepsilon)>0$ and $N$, we obtain
$\delta_{n+1}^{(k+1)}$ $(=\delta)>0$ verifying the estimates of (ii), (iii),
(iv) of Lemma 4.6. Refining the choice of $\delta_{n+1}^{(k)}>0$,
$k\in\mathbb{N}$, making each one smaller in turn, we may suppose that
$\delta_{n+1}^{(k+1)}<\delta_{n+1}^{(k)}$ for $k\in\mathbb{N}$ and

$N\left(\sum_{j=1}^{\infty}8^{j}\delta_{n+1}^{(k+j)}\right)<\delta_{n+1}^{(k)}\quad\text{for
any }k\in\mathbb{N}.$

Since ${\it A}\raisebox{0.23499pt}{\sf F}_{i}\otimes
M_{(2+i)^{\infty}}\cong{\it A}\raisebox{0.23499pt}{\sf F}_{i}$ for $i=0,1$,
there are embeddings
$\epsilon_{n+1}^{(i)}:M_{(2+i)^{L_{n+1}}}\rightarrow({B_{n}^{(i)}})^{\prime}\cap{\it
A}\raisebox{0.23499pt}{\sf F}_{i}$, $i=0,1$, such that

$\displaystyle(\operatorname{Im}(\epsilon_{n+1}^{(1)})\otimes
1_{M_{2^{\infty}}})^{1}$
$\displaystyle\subset_{\delta_{n+1}^{(10)}}\left(\operatorname{Im}(\epsilon_{n+1}^{(0)})\otimes
1_{M_{3^{\infty}}}\right)^{\prime}\cap{\it A}\raisebox{0.23499pt}{\sf F},$
$\displaystyle b\epsilon_{n+1}^{(i)}(1_{M_{(2+i)^{L_{n+1}}}})$
$\displaystyle\approx_{\frac{\varepsilon_{n+1}}{4}}b\quad\text{for any
}i=0,1\text{ and }b\in(B_{n}^{(i)})^{1},\text{ and }$ $\displaystyle
a(\epsilon_{n+1}^{(i)}(1_{M_{(2+i)^{L_{n+1}}}})\otimes
1_{M_{(3-i)^{\infty}}})$
$\displaystyle\approx_{\delta_{n+1}^{(10)}}a\quad\text{ for any }a\in
A_{n}^{1}\cup\bigcup\limits_{j=0,1}\left(B_{n}^{(j)}\otimes
1_{M_{(3-j)^{\infty}}}\right)^{1}.$

By Lemma 4.6 (ii), we obtain a unitary $E_{1}$ in ${\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$ such that
$E_{1}\approx_{\delta_{n+1}^{(9)}}1_{AF}$ and

$\operatorname{Ad}E_{1}(\operatorname{Im}(\epsilon_{n+1}^{(1)})\otimes
1_{M_{2^{\infty}}})\subset\left(\operatorname{Im}(\epsilon_{n+1}^{(0)})\otimes
1_{M_{3^{\infty}}}\right)^{\prime}\cap{\it A}\raisebox{0.23499pt}{\sf F}.$

Set $\overline{\epsilon}_{n+1}(a\otimes b)=\epsilon_{n+1}^{(0)}(a)\otimes
1_{M_{3^{\infty}}}\operatorname{Ad}E_{1}(\epsilon_{n+1}^{(1)}(b)\otimes
1_{M_{2^{\infty}}})$ for $a\in M_{2^{L_{n+1}}}$, $b\in M_{3^{L_{n+1}}}$.
Choosing $\epsilon_{n+1}^{(0)}$ and $\epsilon_{n+1}^{(1)}$ almost commuting
with $\operatorname{Im}(\kappa_{n}^{(i)})^{1}$ and $W_{n}^{(i)}\\!\\!,$ we may
further assume that

$\operatorname{Im}(\kappa_{n}^{(i)})^{1}\cup\\{W_{n}^{(i)}\\}\subset_{\delta_{n+1}^{(8)}}\operatorname{Im}(\overline{\epsilon}_{n+1})^{\prime}\cap{\it
A}\raisebox{0.23499pt}{\sf F},$

for both $i=0,1$. Because of the inclusion
$A_{n}\subset\operatorname{Im}(\kappa_{n}^{(i)})$ for $i=0,1$ in (7, $n$), we
have an embedding $\epsilon_{n+1}:M_{2^{L_{n+1}}}\otimes
M_{3^{L_{n+1}}}\rightarrow A_{n}^{\prime}\cap{\it A}\raisebox{0.23499pt}{\sf
F}$ such that
$\epsilon_{n+1}(x)\approx_{\delta_{n+1}^{(6)}}\overline{\epsilon}_{n+1}(x)$
for all $x\in(M_{2^{L_{n+1}}}\otimes M_{3^{L_{n+1}}})^{1}$. Then it follows
that
$\operatorname{Im}(\epsilon_{n+1})^{1}\subset_{\delta_{n+1}^{(5)}}\operatorname{Im}(\kappa_{n}^{(i)})^{\prime}\cap{\it
A}\raisebox{0.23499pt}{\sf F}$ for $i=0,1$. Note that, since
$1_{A_{n}}\epsilon_{n+1}({1_{M_{2^{L_{n+1}}}\otimes
M_{3^{L_{n+1}}}}})\approx_{\varepsilon_{n+1}}1_{A_{n}}$ and
$\epsilon_{n+1}({1_{M_{2^{L_{n+1}}}\otimes M_{3^{L_{n+1}}}}})\in
A_{n}^{\prime}$, we have $\epsilon_{n+1}({1_{M_{2^{L_{n+1}}}\otimes
M_{3^{L_{n+1}}}}})1_{A_{n}}=1_{A_{n}}$ which implies
$\epsilon_{n+1}({1_{M_{2^{L_{n+1}}}\otimes M_{3^{L_{n+1}}}}})a=a$ for any
$a\in A_{n}$ in (4, $n+1$). For the same reason, from
$1_{A_{n}^{(i)}}\epsilon_{n+1}^{(i)}(1_{M_{(2+i)^{L_{n+1}}}})\approx_{\varepsilon_{n+1}}1_{A_{n}^{(i)}}$
and $\epsilon_{n+1}^{(i)}(1_{M_{(2+i)^{L_{n+1}}}})\in{A_{n}^{(i)}}^{\prime}$,
it follows that $\epsilon_{n+1}^{(i)}(1_{M_{(2+i)^{L_{n+1}}}})a=a$ for all
$i=0,1$ and $a\in A_{n}^{(i)}$.

For $i=0,1$, since ${\it A}\raisebox{0.23499pt}{\sf F}_{i}\otimes
M_{(3-i)^{\infty}}={\it A}\raisebox{0.23499pt}{\sf F}$, there exist
${\widetilde{L}}_{n+1}\in\mathbb{N}$ and a finite dimensional
$\mathrm{C}^{*}$-subalgebra $A_{n+1}^{(i)}$ of ${\it
A}\raisebox{0.23499pt}{\sf F}_{i}$ satisfying $F_{n+1}\otimes
1_{(2+i)^{\infty}}\subset K_{0}(A_{n+1}^{(i)})^{+}$ in (2, $n+1$),
${\widetilde{L}}_{n+1}>L_{n+1}$, $A_{n}^{(i)}\subset A_{n+1}^{(i)}$,
$F_{A_{i}n+1}\subset_{\varepsilon_{n+1}}A_{n+1}^{(i)}$,
$A_{n}\subset_{\delta_{n+1}^{(10)}}A_{n+1}^{(i)}\otimes
M_{(3-i)^{{\widetilde{L}}_{n+1}}}$, and
$B_{n}^{(i)}\operatorname{Im}(\epsilon_{n+1}^{(i)})\subset A_{n+1}^{(i)}$.
When $n$ is an even number (resp. an odd number), we have $i_{n+1}=1$ (resp.
$i_{n+1}=0$). By Lemma 4.6 (ii), there exists ${\widetilde{\delta}}>0$
satisfying the condition for $\delta_{n+1}^{(10)}$ $(=\varepsilon)>0$ and
$\dim(A_{n+1}^{(i_{n+1})})6^{2L_{n+1}}\in\mathbb{N}$. Enlarging the choice of
${\widetilde{L}}_{n+1}\in\mathbb{N}$ and $A_{n+1}^{(1-i_{n+1})}$ further, we
may choose ${\widetilde{L}}_{n+1}$ and $A_{n+1}^{(1-i_{n+1})}$ such that

$\displaystyle(A_{n+1}^{(i_{n+1})}\otimes
M_{(3-i_{n+1})^{L_{n+1}}})^{1}\cup(A_{n}+\mathbb{C}1_{AF})\operatorname{Im}(\epsilon_{n+1})^{1}\cup\operatorname{Im}(\kappa_{n}^{(0)})^{1}\cup\operatorname{Im}(\kappa_{n}^{(1)})^{1}\cup
F_{An+1}$
$\displaystyle\subset_{\min\\{\delta_{n+1}^{(10)},{\widetilde{\delta}}\\}}A_{n+1}^{(1-i_{n+1})}\otimes
M_{(2+i_{n+1})^{{\widetilde{L}}_{n+1}}},\text{ and}$ (1)
$\displaystyle\quad\\{W_{n}^{(0)},W_{n}^{(1)}\\}\subset_{\delta_{n+1}^{(10)}}A_{n+1}^{(1-i_{n+1})}\otimes
M_{(2+i_{n+1})^{{\widetilde{L}}_{n+1}}}+\mathbb{C}1_{AF}.$ (2)

By Lemma 4.6 (ii) for $\delta_{n+1}^{(9)}>0$ and $N\in\mathbb{N}$, there
exists a unitary $U_{n+1}\in{\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$
satisfying the condition of (3, $n+1$) and
$U_{n+1}\approx_{\delta_{n+1}^{(9)}}1_{AF}$. Define

$A_{n+1}=\operatorname{Ad}U_{n+1}(A_{n+1}^{(1-i_{n+1})}\otimes
M_{(2+i_{n+1})^{{\widetilde{L}}_{n+1}}})\subset{\it A}\raisebox{0.23499pt}{\sf
F}.$

Note that from
$F_{An+1}\subset_{\delta_{n+1}^{(10)}}A_{n+1}^{(1-i_{n+1})}\otimes
M_{(2+i_{n+1})^{{\widetilde{L}}_{n+1}}}$, we see that
$F_{An+1}\subset_{\varepsilon_{n+1}}A_{n+1}^{1}$ as required for (4, $n+1$).
By the conditions of (3, $n+1$) and (4, $n+1$), it follows that $A_{n}\subset
A_{n+1}$. Since
$\bigcup_{i=0,1}\operatorname{Im}(\kappa_{n}^{(i)})^{1}\subset_{\delta_{n+1}^{(8)}}A_{n+1}$,
applying Lemma 4.6 (iv) to $A=A_{n+1}$,
$B=\operatorname{Im}(\kappa_{n}^{(i)})$, and $C=A_{n}$, we obtain unitaries
$y_{n}^{(i)}\in{\it A}\raisebox{0.23499pt}{\sf F}^{\sim}\cap A_{n}^{\prime}$,
$i=0,1$, such that
$\operatorname{Ad}y_{n}^{(i)}(\operatorname{Im}(\kappa_{n}^{(i)}))\subset
A_{n+1}$ and $y_{n}^{(i)}\approx_{\delta_{n+1}^{(7)}}1_{AF}$. Define
embeddings $\overline{\kappa}_{n}^{(i)}:B_{n}^{(i)}\otimes
M_{(3-i)^{\Lambda_{n}}}\rightarrow A_{n+1}$, $i=0,1$, by
$\overline{\kappa}_{n}^{(i)}=\operatorname{Ad}y_{n}^{(i)}\circ\kappa_{n}^{(i)}$.
Note that, by (7,$\ n$), we also see that
$\overline{\kappa}_{n}^{(i)}(a\otimes
1_{M_{(3-i)^{\Lambda_{n}}}})=\iota_{n}^{(i)}(a)$ for any $i=0,1$ and $a\in
A_{n}^{(i)}$. Now we have
$A_{n}\subset\operatorname{Im}(\overline{\kappa}_{n}^{(i)})\subset A_{n+1}$
for both $i=0,1$, $\operatorname{Im}(\epsilon_{n+1})\subset A_{n+1}$, and
$\operatorname{Im}(\overline{\kappa}_{n}^{(i)})^{1}\subset_{\delta_{n+1}^{(4)}}A_{n+1}\cap\operatorname{Im}(\epsilon_{n+1})^{\prime}$
for $i=0,1$. Applying Lemma 4.6 (iv) to
$A=A_{n+1}\cap\operatorname{Im}(\epsilon_{n+1})^{\prime}$,
$B=\operatorname{Im}(\overline{\kappa}_{n}^{(i)})$, and $C=A_{n}$, we obtain
unitaries ${\tilde{y}}_{n}^{(i)}\in A_{n}^{\prime}\cap{\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$, $i=0,1$, such that
${\tilde{y}}_{n}^{(i)}\approx_{\delta_{n+1}^{(3)}}1_{AF}$ and
$\operatorname{Ad}{\tilde{y}}_{n}^{(i)}(\operatorname{Im}(\overline{\kappa}_{n}^{(i)}))\subset
A_{n+1}\cap\operatorname{Im}(\epsilon_{n+1})^{\prime}$. Define embeddings
${\widetilde{\kappa}}_{n}^{(i)}:B_{n}^{(i)}\otimes
M_{(3-i)^{\Lambda_{n}}}\rightarrow A_{n+1}$, $i=0,1$, by
${\widetilde{\kappa}}_{n}^{(i)}=\operatorname{Ad}{\tilde{y}}_{n}^{(i)}\circ\overline{\kappa}_{n}^{(i)}$,
which satisfy ${\widetilde{\kappa}}_{n}^{(i)}(a\otimes 1)=\iota_{n}^{(i)}(a)$
for any $i=0,1$ and $a\in A_{n}^{(i)}$, and also
$A_{n}\subset\operatorname{Im}({\widetilde{\kappa}}_{n}^{(i)})\subset
A_{n+1}\cap\operatorname{Im}(\epsilon_{n+1})^{\prime}$ for both $i=0,1$.

To simplify notation, let $\\{e_{ij}\\}_{i,j=1}^{m_{n+1}}$ be a system of
matrix units for $M_{m_{n+1}}\cong M_{2^{L_{n+1}}}\otimes M_{3^{L_{n+1}}}$.
Note that $m_{n+1}>k_{n+1}+l_{n+1}$. Define an embedding $\xi$ from
$C([0,1])\otimes A_{n}$ into $C([0,1])\otimes A_{n+1}$ by

$\xi(f)(t)=\sum_{j=1}^{k_{n+1}}f\left(\frac{t}{2}\right)\epsilon_{n+1}(e_{jj})+\sum_{j=1+k_{n+1}}^{m_{n+1}-l_{n+1}}f\left(\frac{1}{2}\right)\epsilon_{n+1}(e_{jj})+\sum_{j=1+m_{n+1}-l_{n+1}}^{m_{n+1}}f\left(\frac{1+t}{2}\right)\epsilon_{n+1}(e_{jj}),$

for $f\in C([0,1])\otimes A_{n}$ and $t\in[0,1]$. Set
$p=\epsilon_{n+1}\left(\sum_{j=1}^{k_{n+1}}e_{jj}\right)$. Since
${\widetilde{\kappa}}_{n}^{(0)}(a\otimes
1_{M_{3^{\Lambda_{n}}}})=\iota_{n}^{(0)}(a)$ for any $a\in A_{n}^{(0)}$ and
$A_{n}\subset\operatorname{Im}({\widetilde{\kappa}}_{n}^{(0)})$, for any
$f\in\mathscr{A}_{n}$ there exist $a_{f}\in A_{n}^{(0)}\subset B_{n}^{(0)}$
and $b_{f}\in B_{n}^{(0)}\otimes M_{3^{\Lambda_{n}}}$ such that

$\displaystyle\xi(f)(0)$
$\displaystyle=\sum_{j=1}^{k_{n+1}}f(0)\epsilon_{n+1}(e_{jj})+\sum_{j=1+k_{n+1}}^{m_{n+1}}f\left(\frac{1}{2}\right)\epsilon_{n+1}(e_{jj})$
$\displaystyle={\widetilde{\kappa}}_{n}^{(0)}(a_{f}\otimes
1)p+{\widetilde{\kappa}}_{n}^{(0)}(b_{f})(\epsilon_{n+1}(1)-p).$

Since $3^{L_{n+1}}|3^{L_{n+1}}k_{n+1}^{\prime}=k_{n+1}$ and
$3^{\Lambda_{n}}|2^{L_{n+1}}-k_{n+1}^{\prime}$, applying Lemma 4.6 (i) to
$P=3^{\Lambda_{n}}$, $Q=2^{L_{n+1}}$, $R=3^{L_{n+1}}$, and $p$, we obtain a
unitary $U_{n+1}^{(0)}$ in
${\widetilde{\kappa}}_{n}^{(0)}(1_{B_{n}^{(0)}}\otimes
M_{3^{\Lambda_{n}}})\epsilon_{n+1}(M_{2^{L_{n+1}}}\otimes
M_{3^{L_{n+1}}})+\mathbb{C}1_{AF}\subset A_{n+1}+\mathbb{C}1_{AF}$ such that

$\operatorname{Ad}U_{n+1}^{(0)}(\xi(f)(0))\in{\widetilde{\kappa}}_{n}^{(0)}(B_{n}^{(0)}\otimes
1_{M_{3^{\Lambda_{n}}}})\epsilon_{n+1}(M_{2^{L_{n+1}}}\otimes
1_{M_{3^{L_{n+1}}}}),$

for any $f\in\mathscr{A}_{n}$. By (6, $n$), now we have
$\operatorname{Ad}W_{n}^{(0)}(b\otimes
1_{M_{3^{\infty}}})=\kappa_{n}^{(0)}(b\otimes 1_{M_{3^{\Lambda_{n}}}})$ for
any $\in B_{n}^{(0)}$ and
$W_{n}^{(0)}\in_{\varepsilon_{n}/4}\operatorname{Im}(\kappa_{n}^{(0)})+\mathbb{C}1_{AF}$.
Since
$W_{n}^{(0)}\in_{\delta_{n+1}^{(8)}}\operatorname{Im}(\overline{\epsilon}_{n+1})^{\prime}\cap{\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$ (from (2)), and
$\overline{\epsilon}_{n+1}(x)\approx_{\delta_{n+1}^{(6)}}\epsilon_{n+1}(x)$
for all $x\in(M_{2^{L_{n+1}}}\otimes M_{3^{L_{n+1}}})^{1}$, we obtain a
unitary
${\widetilde{W}}_{n}^{(0)}\in\operatorname{Im}(\epsilon_{n+1})^{\prime}\cap{\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$ such that
${\widetilde{W}}_{n}^{(0)}\approx_{\delta_{n+1}^{(5)}}W_{n}^{(0)}$. Then it
follows that

$\displaystyle\operatorname{Ad}{W_{n}^{(0)}}^{*}\left({\widetilde{\kappa}}_{n}^{(0)}(b\otimes
1_{M_{3^{\Lambda_{n}}}})\epsilon_{n+1}(a\otimes 1_{M_{3^{L_{n+1}}}})\right)$
$\displaystyle\approx_{2\delta_{n+1}^{(5)}}\operatorname{Ad}{W_{n}^{(0)}}^{*}\left({\widetilde{\kappa}}_{n}^{(0)}(b\otimes
1_{M_{3^{\Lambda_{n}}}})\right)\epsilon_{n+1}(a\otimes 1_{M_{3^{L_{n+1}}}})$
$\displaystyle=(b\otimes 1_{M_{3^{\infty}}})\epsilon_{n+1}(a\otimes
1_{M_{3^{L_{n+1}}}})\approx_{\delta_{n+1}^{(6)}+2\delta_{n+1}^{(9)}}((b\epsilon_{n+1}^{(0)}(a))\otimes
1_{M_{3^{\infty}}})(\epsilon_{n+1}^{(1)}(1)\otimes 1_{M_{2^{\infty}}})$
$\displaystyle\approx_{\delta_{n+1}^{(10)}}b\epsilon_{n+1}^{(0)}(a)\otimes
1_{M_{3^{\infty}}},$

for any $b\in{B_{n}^{(0)}}^{1}$ and $a\in M_{2^{L_{n+1}}}^{1}$. By Lemma 4.6
(iii), we obtain a unitary $u^{(0)}$ in ${\it A}\raisebox{0.23499pt}{\sf
F}^{\sim}$ such that $u^{(0)}\approx_{\delta_{n+1}^{(3)}}1_{AF}$ and

$\operatorname{Ad}u^{(0)}{W_{n}^{(0)}}^{*}\left({\widetilde{\kappa}}_{n}^{(0)}(b\otimes
1_{M_{3^{\Lambda_{n}}}})\epsilon_{n+1}(a\otimes
1_{M_{3^{L_{n+1}}}})\right)=(b\epsilon_{n+1}^{(0)}(a))\otimes
1_{M_{3^{\infty}}},$

for any $b\in B_{n}^{(0)}$ and $a\in M_{2^{L_{n+1}}}$. Consider the unitary
$z_{n}^{(0)}=W_{n}^{(0)}{u^{(0)}}^{*}$ in ${\it A}\raisebox{0.23499pt}{\sf
F}^{\sim}$, which satisfies
$z_{n}^{(0)}\approx_{\delta_{n+1}^{(3)}}W_{n}^{(0)}$ and
$\operatorname{Ad}z_{n}^{(0)}(b\epsilon_{n+1}^{(0)}(a)\otimes
1_{M_{3^{\infty}}})={\widetilde{\kappa}}_{n}^{(0)}(b\otimes
1_{M_{3^{\Lambda_{n}}}})\epsilon_{n+1}(a\otimes 1_{M_{3^{L_{n+1}}}})$ for any
$b\in B_{n}^{(0)}$ and $a\in M_{2^{L_{n+1}}}$. Define an embedding
${\overline{\iota}}_{n+1}^{(0)}$ of $A_{n+1}^{(0)}$ into ${\it
A}\raisebox{0.23499pt}{\sf F}$ by

${\overline{\iota}}_{n+1}^{(0)}(a)=\operatorname{Ad}{U_{n+1}^{(0)}}^{*}z_{n}^{(0)}(a\otimes
1_{M_{3^{\infty}}})\quad\text{for }a\in A_{n+1}^{(0)}.$

From $B_{n}^{(0)}\operatorname{Im}(\epsilon_{n+1}^{(0)})\subset
A_{n+1}^{(0)}$, it follows that
$\xi(f)(0)\in\operatorname{Im}({\overline{\iota}}_{n+1}^{(0)})$ for any
$f\in\mathscr{A}_{n}$.

In the same way as for $i=0$, we also obtain unitaries $u^{(1)}$,
$U_{n+1}^{(1)}$, $z_{n}^{(1)}$ in ${\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$,
and an embedding ${\overline{\iota}}_{n+1}^{(1)}$ of $A_{n+1}^{(1)}$ into
${\it A}\raisebox{0.23499pt}{\sf F}$ such that
$U_{n+1}^{(1)}\in{\widetilde{\kappa}}_{n}^{(1)}(1_{B_{n}^{(1)}}\otimes
M_{2^{\Lambda_{n}}})\epsilon_{n+1}(M_{2^{L_{n+1}}}\otimes
M_{3^{L_{n+1}}})+\mathbb{C}1_{AF}$,
$z_{n}^{(1)}=W_{n}^{(1)}{u^{(1)}}^{*}\approx_{\delta_{n+1}^{(3)}}W_{n}^{(1)}$,
${\overline{\iota}}_{n+1}^{(1)}(a)=\operatorname{Ad}{U_{n+1}^{(1)}}^{*}z_{n}^{(1)}(a\otimes
1_{M_{2^{\infty}}})$ for $a\in A_{n+1}^{(1)}$, and
$\xi(f)(1)\in\operatorname{Im}({\overline{\iota}}_{n+1}^{(1)})$ for any
$f\in\mathscr{A}_{n}$.

Note that, since $(A_{n+1}^{(i)}\otimes
1_{M_{(3-i)^{L_{n+1}}}})^{1}\subset_{\delta_{n+1}^{(8)}}A_{n+1}$ for both
$i=0,1$, and since
$z_{n}^{(i)}\approx_{\delta_{n+1}^{(3)}}W_{n}^{(i)}\in_{\delta_{n+1}^{(8)}}A_{n+1}+\mathbb{C}1_{AF}$
(from (2)), we have

$\operatorname{Im}({\overline{\iota}}_{n+1}^{(i)})^{1}=\operatorname{Ad}{U_{n+1}^{(i)}}^{*}z_{n}^{(i)}((A_{n+1}^{(i)})^{1}\otimes
1_{M_{(3-i)^{L_{n+1}}}})\subset_{\delta_{n+1}^{(2)}}A_{n+1}.$

Although $\dim(A_{n+1}^{(i)})$ can be bigger than $N$, the following argument
allows us to obtain unitaries ${\widetilde{U}}_{n+1}^{(i)}\in{\it
A}\raisebox{0.23499pt}{\sf F}^{\sim}$, $i=0,1$, such that
${\widetilde{U}}_{n+1}^{(i)}\approx_{\delta_{n+1}^{(2)}}1_{AF}$ and
$\operatorname{Ad}{\widetilde{U}}_{n+1}^{(i)}\circ{\overline{\iota}}_{n+1}^{(i)}(A_{n+1}^{(i)})\subset
A_{n+1}$. In the case $i=1-i_{n+1}$, we fix a unitary
${\overline{U}}_{n+1}^{(i)}=$ $U_{n+1}u^{(i)}{W_{n}^{(i)}}^{*}$
$U_{n+1}^{(i)}\in{\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$. Since
$W_{n}^{(i)}\in_{\delta_{n+1}^{(8)}}A_{n+1}+\mathbb{C}1_{AF}$ and
$U_{n+1}^{(i)}\in A_{n+1}+\mathbb{C}1_{AF}$, there exists a unitary
${\overline{y}}^{(i)}\in A_{n+1}+\mathbb{C}1_{AF}$ such that
${\overline{y}}^{(i)}\approx_{\delta_{n+1}^{(2)}}{\overline{U}}_{n+1}^{(i)}$.
Consider the unitary
${\widetilde{U}}_{n+1}^{(i)}={{\overline{y}}^{(i)}}^{*}{\overline{U}}_{n+1}^{(i)}$,
which satisfies
${\widetilde{U}}_{n+1}^{(i)}\approx_{\delta_{n+1}^{(2)}}1_{AF}$ and
$\operatorname{Ad}{\widetilde{U}}_{n+1}^{(i)}\circ{\overline{\iota}}_{n+1}^{(i)}(a)=\operatorname{Ad}{{\overline{y}}^{(i)}}^{*}U_{n+1}(a\otimes
1_{M_{(3-i)^{\infty}}})\in A_{n+1}$ for any $a\in A_{n+1}^{(i)}$. In the case
$i=i_{n+1}$, from (1) it follows that $(A_{n+1}^{(i)}\otimes
M_{(3-i)^{L_{n+1}}})^{1}\subset_{{\widetilde{\delta}}}A_{n+1}^{(1-i)}\otimes
M_{(2+i)^{{\widetilde{L}}_{n+1}}}$ and ${\widetilde{\delta}}>0$ is chosen for
$\dim(A_{n+1}^{(i)})6^{2L_{n+1}}$. By Lemma 4.6 (ii), there exists a unitary
$V\in{\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$ such that
$V\approx_{\delta_{n+1}^{(10)}}1_{AF}$ and
$\operatorname{Ad}V(A_{n+1}^{(i)}\otimes M_{(3-i)^{L_{n+1}}})\subset
A_{n+1}^{(1-i)}\otimes M_{(2+i)^{{\widetilde{L}}_{n+1}}}$. It follows that
$\operatorname{Ad}U_{n+1}V(A_{n+1}^{(i)}\otimes M_{(3-i)^{L_{n+1}}})\subset
A_{n+1}$. Set
${\overline{U}}_{n+1}^{(i)}=U_{n+1}Vu^{(i)}{W_{n}^{(i)}}^{*}U_{n+1}^{(i)}$
which is a unitary in ${\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$ such that
${\overline{U}}_{n+1}^{(i)}\in_{2\delta_{n+1}^{(3)}}A_{n+1}+\mathbb{C}1_{AF}$.
Then there exists a unitary ${\overline{y}}^{(i)}\in A_{n+1}+\mathbb{C}1_{AF}$
such that
${\overline{y}}^{(i)}\approx_{\delta_{n+1}^{(2)}}{\overline{U}}_{n+1}^{(i)}$.
Setting
${\widetilde{U}}_{n+1}^{(i)}={{\overline{y}}^{(i)}}^{*}{\overline{U}}_{n+1}^{(i)}$,
we also see that
$\operatorname{Ad}{\widetilde{U}}_{n+1}^{(i)}\circ{\overline{\iota}}_{n+1}^{(i)}(a)=\operatorname{Ad}{{\overline{y}}^{(i)}}^{*}U_{n+1}V(a\otimes
1_{M_{(3-i)^{\infty}}})\in A_{n+1}$ for any $a\in A_{n+1}^{(i)}$.

Define embeddings $\iota_{n+1}^{(i)}$ of $A_{n+1}^{(i)}$ into $A_{n+1}$,
$i=0,1$, by
$\iota_{n+1}^{(i)}=\operatorname{Ad}{\widetilde{U}}_{n+1}^{(i)}\circ{\overline{\iota}}_{n+1}^{(i)}$.
Then it follows that
$\operatorname{Ad}{\widetilde{U}}_{n+1}^{(i)}(\xi(f)(i))\in\operatorname{Im}(\iota_{n+1}^{(i)})$
for any $i=0,1$ and $f\in\mathscr{A}_{n}$. Define unitaries
$V_{n+1}^{(i)}={\widetilde{U}}_{n+1}^{(i)}{U_{n+1}^{(i)}}^{*}z_{n}^{(i)}$,
$i=0,1$, in ${\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$. Since
$U_{n+1}^{(i)}\in A_{n+1}+\mathbb{C}1_{AF}$ for both $i=0,1$, we see that
$V_{n+1}^{(i)}$, $i=0,1$, satisfy the conditions of (6, $n+1$). We define a
$\mathrm{C}^{*}$-subalgebra $\mathscr{A}_{n+1}$ of $C([0,1])\otimes A_{n+1}$
as required for (5, $n+1$). Define finite dimensional
$\mathrm{C}^{*}$-subalgebras $E^{(i)}$, $i=0,1$, of $A_{n+1}$ by
$E^{(i)}=\\{\xi(f)(i)\ :\ f\in\mathscr{A}_{n}\\}$. Note that
$\dim(E^{(i)})\leq\dim(A_{n}^{(i)})+\dim(B_{n}^{(i)})(3-i)^{2\Lambda_{n}}<N$
for both $i=0,1$ and that
$\xi(f)(i)\approx_{2\delta_{n+1}^{(2)}}\operatorname{Ad}{\widetilde{U}}_{n+1}^{(i)}(\xi(f)(i))$
for any $f\in\mathscr{A}_{n}^{1}$. By Lemma 4.6 (iii) with (3, $n+1$), we
obtain unitaries $\mathcal{U}^{(i)}$ in $A_{n+1}+\mathbb{C}1_{AF}$, $i=0,1$,
such that $\mathcal{U}^{(i)}\approx_{\delta_{n+1}^{(0)}}1_{AF}$ and
$\operatorname{Ad}\mathcal{U}^{(i)}(\xi(f)(i))=\operatorname{Ad}{\widetilde{U}}_{n+1}^{(i)}(\xi(f)(i))$
for any $i=0,1$ and $f\in\mathscr{A}_{n}$. Thus we obtain a unitary
$\widetilde{\mathcal{U}}\in C([0,1])\otimes(A_{n+1}+\mathbb{C}1_{AF})$ which
is a Lipschitz continuous function such that
$\widetilde{\mathcal{U}}(i)=\mathcal{U}^{(i)}$ for $i=0,1$,
$\widetilde{\mathcal{U}}(t)\approx_{\varepsilon_{n+1}}1_{AF}$ for all
$t\in[0,1]$ and
$\operatorname{Lip}(\widetilde{\mathcal{U}})\leq\varepsilon_{n+1}$. We define
an embedding $\varphi_{n+1}$ of $\mathscr{A}_{n}$ into $\mathscr{A}_{n+1}$ by
$\varphi_{n+1}(f)=\operatorname{Ad}\widetilde{\mathcal{U}}\circ\xi(f)$ for
$f\in\mathscr{A}_{n}$. Because of the construction of $\xi$ and the inequality
$\operatorname{Lip}(\widetilde{\mathcal{U}})\leq\varepsilon_{n+1}$, it follows
that $\varphi_{n+1}$ satisfies (8, $n+1$).

In the rest of the induction, we construct $\Lambda_{n+1}$, $B_{n+1}^{(i)}$
and $\kappa_{n+1}^{(i)}$ satisfying the conditions (1, $n+1$), (2, $n+1$), (6,
$n+1$), (7, $n+1$). Set ${\overline{\delta}}_{n+1}^{(0)}=\varepsilon_{n+1}/8$.
By applying Lemma 4.6 (ii), (iii), (iv) to ${\overline{\delta}}_{n+1}^{(k)}$
$(=\varepsilon)$ and ${\overline{N}}=\dim(A_{n+1})$ inductively, there exists
${\overline{\delta}}_{n+1}^{(k+1)}$ $(=\delta)>0$ satisfying the conditions of
(ii), (iii), and (iv). Taking smaller ${\overline{\delta}}_{n+1}^{(k)}$,
$k\in\mathbb{N}$, as above, we may assume that
${\overline{\delta}}_{n+1}^{(k+1)}<{\overline{\delta}}_{n+1}^{(k)}$ for
$k\in\mathbb{N}$ and
$N\sum_{j=1}^{\infty}8^{j}{\overline{\delta}}_{n+1}^{(k+j)}<{\overline{\delta}}_{n+1}^{(k)}$.
Since ${\it A}\raisebox{0.23499pt}{\sf F}={\it A}\raisebox{0.23499pt}{\sf
F}_{i}\otimes M_{(3-i)^{\infty}}$ for both $i=0,1$, there exist
$\Lambda_{n+1}\in\mathbb{N}$ with the condition of (1, $n+1$) and finite
dimensional $\mathrm{C}^{*}$-subalgebras $B_{n+1}^{(i)}\subset{\it
A}\raisebox{0.23499pt}{\sf F}_{i}$, $i=0,1$, such that $A_{n+1}^{(i)}\subset
B_{n+1}^{(i)}$,
$A_{n+1}^{1}\subset_{{\overline{\delta}}_{n+1}^{(4)}}B_{n+1}^{(i)}\otimes
M_{(3-i)^{\Lambda_{n+1}}}$, and
$\\{V_{n+1}^{(0)},V_{n+1}^{(1)}\\}\subset_{{\overline{\delta}}_{n+1}^{(4)}}B_{n+1}^{(i)}\otimes
M_{(3-i)^{\Lambda_{n+1}}}+\mathbb{C}1_{AF}$.

By Lemma 4.6 (ii), there exist unitaries ${\overline{W}}_{n+1}^{(i)}$,
$i=0,1$, in ${\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$ such that
${\overline{W}}_{n+1}^{(i)}\approx_{{\overline{\delta}}_{n+1}^{(3)}}1_{AF}$
and $\operatorname{Ad}{\overline{W}}_{n+1}^{(i)}(B_{n+1}^{(i)}\otimes
M_{(3-i)^{\Lambda_{n+1}}})\supset A_{n+1}$ for $i=0,1$. Define embeddings
$\widehat{\kappa}_{n+1}^{(i)}:B_{n+1}^{(i)}\otimes
M_{(3-i)^{\Lambda_{n+1}}}\rightarrow{\it A}\raisebox{0.23499pt}{\sf F}$,
$i=0,1$, by
$\widehat{\kappa}_{n+1}^{(i)}=\operatorname{Ad}{\overline{W}}_{n+1}^{(i)}$. By
Lemma 4.6 (v) and
$V_{n+1}^{(i)}\in_{{\overline{\delta}}_{n+1}^{(2)}}\operatorname{Im}(\widehat{\kappa}_{n+1}^{(i)})+\mathbb{C}1_{AF}$,
there exist unitaries
$\widehat{W}_{n+1}^{(i)}\in\operatorname{Im}(\widehat{\kappa}_{n+1}^{(i)})+\mathbb{C}1_{AF}$
such that $\operatorname{Ad}\widehat{W}_{n+1}^{(i)}V_{n+1}^{(i)}(a\otimes
1_{M_{(3-i)^{\infty}}})=\operatorname{Ad}{\overline{W}}_{n+1}^{(i)}(a\otimes
1_{M_{(3-i)^{\infty}}})$ for any $a\in A_{n+1}^{(i)}$. Define unitaries
$W_{n+1}^{(i)}$, $i=0,1$, in ${\it A}\raisebox{0.23499pt}{\sf F}^{\sim}$ and
embeddings $\kappa_{n+1}^{(i)}:B_{n+1}^{(i)}\otimes
M_{(3-i)^{\Lambda_{n+1}}}\rightarrow{\it A}\raisebox{0.23499pt}{\sf F}$ by

$W_{n+1}^{(i)}=(\widehat{W}_{n+1}^{(i)})^{*}{\overline{W}}_{n+1}^{(i)},\quad\kappa_{n+1}^{(i)}(b\otimes
a)=\operatorname{Ad}W^{(i)}(b\otimes a),$

for $b\in B_{n+1}^{(i)}$ and $a\in M_{(3-i)^{\Lambda_{n+1}}}\subset
M_{(3-i)^{\infty}}$. These choices satisfy the conditions of (6, $n+1$) and
(7, $n+1$).

Define $A_{G}$ as the inductive limit $\mathrm{C}^{*}$-algebra
$\displaystyle\lim_{\longrightarrow}(\mathscr{A}_{n},\varphi_{n+1})$. In the
proof of Proposition 4.5 (iii), we have seen that
$(G,G^{+})\cong(\operatorname{Im}(\iota_{\infty*}^{(0)})\cap\operatorname{Im}(\iota_{\infty*}^{(1)}),\operatorname{Im}(\iota_{\infty*}^{(0)})\cap\operatorname{Im}(\iota_{\infty*}^{(1)})\cap
K_{0}({\it A}\raisebox{0.23499pt}{\sf F})^{+})$, as ordered abelian groups. To
show that the right hand side is isomorphic to
$(K_{0}(A_{G}),K_{0}(A_{G})^{+})$, we need the following observation. Let
$\eta_{n}$ denote the canonical embedding of $A_{n}$ into $A_{n+1}$ which is
obtained from (3, $n+1$) for $n\in\mathbb{N}$. Write
$\operatorname{ev}_{n}^{(i)}:\mathscr{A}_{n}\rightarrow\operatorname{Im}(\iota_{n}^{(i)})$,
$i=0,1$, to denote the evaluation maps and
${\overline{\eta}}_{n}^{(i)}:\operatorname{Im}(\iota_{n}^{(i)})\rightarrow
A_{n}$ for the canonical embeddings.

######  Observation 4.7.

For $i=0,1$ and $n\in\mathbb{N}$, the two $*$-homomorphisms
$\operatorname{ev}_{n+1}^{(i)}\circ\varphi_{n+1}$ and
$\operatorname{Ad}\widetilde{\mathcal{U}}(i)\circ\eta_{n}\circ{\overline{\eta}}_{n}^{(i)}\circ\operatorname{ev}_{n}^{(i)}$
$:\mathscr{A}_{n}\rightarrow\operatorname{Im}(\iota_{n+1}^{(i)})$ are
homotopic.

###### Proof.

Let $\Phi_{t}^{(i)}$, $t\in[0,1]$, $i=0,1$, be pointwise continuous paths of
automorphisms on $\mathscr{A}_{n}$ such that
$\Phi_{0}^{(0)}=\Phi_{0}^{(1)}=\operatorname{id}_{\mathscr{A}_{n}}$,
$\Phi_{1}^{(0)}(f)(t)=f(0)$, $t\in[0,1/2]$, $\Phi_{1}^{(0)}(f)(t)=f(2t-1)$,
$t\in[1/2,1]$, $\Phi_{1}^{(1)}(f)(t)=f(2t)$, $t\in[0,1/2]$, and
$\Phi_{1}^{(1)}(f)(t)=f(1)$, $t\in[1/2,1]$, for any $f\in\mathscr{A}_{n}$.
Then it follows that $\operatorname{ev}_{n+1}^{(i)}\circ\varphi_{n+1}$ and
$\operatorname{ev}_{n+1}^{(i)}\circ\varphi_{n+1}\circ\Phi_{1}^{(i)}$ are
homotopic for both $i=0,1$. From the construction of
$\varphi_{n+1}=\operatorname{Ad}\widetilde{\mathcal{U}}\circ\xi$ and
$\Phi_{1}^{(i)}(f)(1/2)=f(i)$, we have

$\operatorname{ev}_{n+1}^{(i)}\circ\varphi_{n+1}\circ\Phi_{1}^{(i)}(f)=\operatorname{Ad}\widetilde{\mathcal{U}}(i)(f(i)\epsilon_{n+1}(1_{M_{m_{n+1}}}))=\operatorname{Ad}\widetilde{\mathcal{U}}(i)\circ\eta_{n}\circ{\overline{\eta}}_{n}^{(i)}\circ\operatorname{ev}_{n}^{(i)}(f),$

for any $i=0,1$ and $f\in\mathscr{A}_{n}$. ∎

Denote by $\eta_{\infty n}$ the canonical embedding of $A_{n}$ into ${\it
A}\raisebox{0.23499pt}{\sf F}$. For $i=0,1$, denote by
$\operatorname{Ev}_{n}^{(i)}:\mathscr{A}_{n}\rightarrow A_{n}$ the evaluation
map at $i$. Note that
${\overline{\eta}}_{n}^{(i)}\circ\operatorname{ev}_{n}^{(i)}=\operatorname{Ev}_{n}^{(i)}$.
Let $\eta_{n}^{(i)}:A_{n}^{(i)}\rightarrow A_{n+1}^{(i)}$ denote the canonical
embeddings of (2, $n+1$) and $\eta_{\infty n}^{(i)}:A_{n}^{(i)}\rightarrow{\it
A}\raisebox{0.23499pt}{\sf F}_{i}$ the canonical embeddings. For $m$,
$n\in\mathbb{N}$ with $m>n$, set
$\varphi_{m,n}=\varphi_{m}\circ\varphi_{m-1}\circ\cdots\circ\varphi_{n+1}:\mathscr{A}_{n}\rightarrow\mathscr{A}_{m}$,
$\eta_{m,n}=\eta_{m-1}\circ\eta_{m-2}\circ\cdots\circ\eta_{n}:A_{n}\rightarrow
A_{m}$, and
$\eta_{m,n}^{(i)}=\eta_{m-1}^{(i)}\circ\eta_{m-2}^{(i)}\circ\cdots\circ\eta_{n}^{(i)}:A_{n}^{(i)}\rightarrow
A_{m}^{(i)}$. By Observation 4.7,
$\operatorname{Ev}_{n+1}^{(i)}\circ\varphi_{n+1}$ and
$\operatorname{Ad}\widetilde{\mathcal{U}}(i)\circ\eta_{n}\circ\operatorname{Ev}_{n}^{(i)}$
are homotopic as $*$-homomorphisms from $\mathscr{A}_{n}$ into $A_{n+1}$.
Since $\widetilde{\mathcal{U}}(i)\in A_{n+1}+\mathbb{C}1_{AF}$, it follows
that
$\operatorname{Ev}_{n+1*}^{(i)}\circ\varphi_{n+1*}=\eta_{n*}\circ\operatorname{Ev}_{n*}^{(i)}$.
From (6, $n$) it follows that

$\iota_{n+1}^{(i)}\circ\eta_{n}^{(i)}=\operatorname{Ad}V_{n+1}^{(i)}{V_{n}^{(i)}}^{*}\circ\iota_{n}^{(i)},\text{
and
}V_{n+1}^{(i)}{V_{n}^{(i)}}^{*}\in_{2\varepsilon_{n}}A_{n+1}+\mathbb{C}1_{AF}.$

Then we have