# Tracial oscillation zero and ${\cal Z}$-stability

Huaxin Lin

###### Abstract

Let $A$ be a (not necessarily unital) separable non-elementary simple amenable
$C^{*}$-algebra whose tracial basis may not have finite covering dimension and
may not be compact but satisfies certain condition (C). We show that $A$ is
${\cal Z}$-stable if and only if $A$ has strict comparison for positive
elements. Extremal boundaries of simplexes which satisfy condition (C) may
contain countable disjoint unions of $n$-dimensional cubes ($n\in\mathbb{N}$)
as a subset.

##  1 Introduction

The recent progress in the classification of simple $C^{*}$-algebras are for
separable simple amenable $C^{*}$-algebras which tensorially absorb the Jiang-
Su algebra. These $C^{*}$-algebras are called ${\cal Z}$-stable
$C^{*}$-algebras. In fact, almost all classification results in the Elliott
program, the program of classification of separable amenable $C^{*}$-algebras,
are for ${\cal Z}$-stable $C^{*}$-algebras ([17], [18], [49], [46], [34],
[19], [20], [40], [42], [28], [21], [56], to name only a few). The Jiang-Su
algebra ${\cal Z}$ is an infinite dimensional simple $C^{*}$-algebra with a
unique tracial state whose ordered $K$-theory is exactly the same as that of
the complex field (see [33]). If $A$ is a separable simple $C^{*}$-algebra
with weakly unperforated $K_{0}(A),$ then $A$ and $A\otimes{\cal Z}$ have
exactly the same Elliott invariant (see [27]). In other words, the current
restriction to ${\cal Z}$-stable $C^{*}$-algebras in the Elliott program is
not the limitation of methods but rather a discovery.

The fact that regularity conditions are required in the study of the
classification program has a long history. It was in the horizon since B.
Blackadar’s strict comparison was introduced in 1980’s (see [4]). Perhaps, the
first deep result in regularity conditions for simple $C^{*}$-algebras is the
introduction of the notion of slow dimension growth in [15]. Stable rank
([47]), real rank ([8]), tracial rank ([39]), decomposition rank ([37]), as
well as nuclear dimension ([60]) have been introduced to the $C^{*}$-algebra
theory. They may all be regarded as regularity conditions.

Toms-Winter’s conjecture states (for stably finite case) that for a separable
non-elementary stably finite simple amenable $C^{*}$-algebra $A,$ the
following are equivalent:

(1) $A$ has strict comparison,

(2) $A\cong A\otimes{\cal Z},$

(3) $A$ has finite nuclear dimension.

By now, the equivalence of (2) and (3) has been established (see [10], [9],
[59], [54], [44]). The implication (2) $\Rightarrow$ (1) is established
earlier ([50]).

The remaining direction is the implication (1) $\Rightarrow$ (2) (or (3)). A
breakthrough was made by the celebrated work of Matui and Sato ([43]) which
shows, among other directions of Toms-Winter’s conjecture, that, if $A$ is a
unital stably finite separable non-elementary amenable simple $C^{*}$-algebra
with finitely many extremal traces and has strict comparison, then $A\cong
A\otimes{\cal Z}.$ The newly invented methods was later developed further (see
[35], [52] and [58]) to show that Toms-Winter’s conjecture holds for those
unital separable simple amenable $C^{*}$-algebras $A$ whose tracial state
space $T(A)$ is a Bauer simplex with finite dimensional extremal boundary. Wei
Zhang ([61]) showed that Toms-Winter conjecture holds for unital separable
amenable simple $C^{*}$-algebras whose extremal tracial states are finite-
dimensional with tightness property (but not closed). More recently in [11],
it is shown that Toms-Winter’s conjecture holds under the assumption of
uniform property $\Gamma.$

In this paper, we show that the conjecture holds under the additional
conditions that $A$ has stable rank one and the tracial cone
$\widetilde{T}(A)$ (the cone of densely defined traces) has a basis with
condition (C) (see Definition 2.12 and Theorem 5.6). If one of the basis has
countable extremal boundary, then the condition that $A$ has stable rank one
can be dropped (see Corollary 5.7). One example of Choquet simplex $T$ which
satisfies the condition (C) is the simplex whose extremal boundary
$\partial_{e}(T)$ is the one point compactification of countable disjoint
union of finite (but not bounded) dimensional cubes (or any metric spaces with
finite covering dimension). So $\partial_{e}(T)$ is not finite dimensional.
There are also examples of Choquet simplexes which satisfy condition (C) but
are not Bauer simplexes and their extremal boundaries are not finite
dimensional (see examples 2.13). It should be noted that extremal boundaries
$\partial_{e}(T(A))$ of $T(A)$ in [43], [35], [52] and [58]), as well as in
[61] are all finite dimensional. Except those studied in [61], tracial state
spaces $T(A)$ studied in these papers ([43], [35], [52] and [58]) all satisfy
condition (C).

If $A$ is not unital but $A\otimes{\cal K}$ has a nonzero projection $p,$ then
the previous results apply to $C^{*}$-algebras $B=p(A\otimes{\cal K})p$ which
are unital. Since $A$ is assumed to be separable and simple, by Brown’s stable
isomorphism theorem (see [6]), one obtains $B\otimes{\cal K}\cong
A\otimes{\cal K}.$ Therefore the results mentioned in [43], [35], [52] and
[58] can be applied to the case that $A$ is not unital but
$K_{0}(A)_{+}\not=\\{0\\}.$ However, this stabilization method fails when $A$
is a stably projectionless $C^{*}$-algebra. Theorem 5.6 in this paper includes
the case that $A$ is a stably projectionless separable simple amenable
$C^{*}$-algebra whose $\widetilde{T}(A)$ has a basis which satisfies condition
(C) (see 2.12 and also 2.17).

This paper uses the notion of tracial approximate oscillation zero. Let us,
for simplicity, assume that $A$ is a unital separable simple $C^{*}$-algebra.
Denote by $\Gamma:{\rm Cu}(A)\to{\rm LAff}_{+}(T(A))$ (the set of strictly
positive lower-semicontinuous affine functions on the tracial state space
$T(A)$) the canonical map defined by $\langle a\rangle\mapsto\widehat{\langle
a\rangle}=d_{\tau}(a)$ (the dimension function of $a$). In general
$\widehat{\langle a\rangle}$ is not continuous. Let $\omega(a)$ be the
oscillation of the lower-semicontinuous function $\widehat{\langle a\rangle}.$
Roughly speaking, a separable simple $C^{*}$-algebra has tracial approximate
oscillation zero, if every positive element in $A$ can be approximated
(tracially) by those elements whose tracial oscillations tend to zero. This
notion will be discussed in detail in [26]. Among other things, we show that
if $A$ is a (not necessarily unital) separable simple $C^{*}$-algebra which
has strict comparison, then $A$ has tracial approximate oscillation zero if
and only if $A$ has stable rank one and the canonical map $\Gamma$ is
surjective. In the case that $\partial_{e}(T(A))$ has countably many points,
$A$ always has tracial approximate oscillation zero. Perhaps, the notion of
tracial oscillation zero may play further role in the study of simple
$C^{*}$-algebras.

The paper is organized as follows. Section 2 serves as the preliminary of the
paper, where, among other items, we provide examples of non-Bauer simplexes
which satisfy condition (C). Some of them are not finite dimensional. We also
show that, for non-unital separable simple $C^{*}$-algebras, the condition
that one of its bases for the cone of traces satisfies condition (C) does not
depend on the choice of the basis. In section 3 we recall some variations of
results related to Matui-Sato’s paper [43] and we mix them with the notion of
tracial approximate oscillation zero. In section 4, we present some technical
results which allow us to add maps approximately orthogonally, using the
notion of tracial oscillation zero. The last section contains the proof of the
main result of the paper.

Acknowlegements This research is partially supported by a NSF grant (DMS
1954600) and the Research Center for Operator Algebras in East China Normal
University which is partially supported by Shanghai Key Laboratory of PMMP,
Science and Technology Commission of Shanghai Municipality (STCSM), grant
#13dz2260400.

##  2 Preliminary

######  Definition 2.1.

Let $A$ be a $C^{*}$-algebra and $S\subset A$ a subset of $A.$ Denote by ${\rm
Her}(S)$ the hereditary $C^{*}$-subalgebra of $A$ generated by $S.$ Denote by
$A^{\bf 1}$ the closed unit ball of $A,$ and by $A_{+}$ the set of all
positive elements in $A.$ Put $A_{+}^{\bf 1}:=A_{+}\cap A^{\bf 1}.$ Denote by
$\widetilde{A}$ the minimal unitization of $A.$ Let ${\rm Ped}(A)$ denote the
Pedersen ideal of $A$ and ${\rm Ped}(A)_{+}:={\rm Ped}(A)\cap A_{+}$. Denote
by $T(A)$ the tracial state space of $A.$

######  Definition 2.2.

Let $A$ and $B$ be $C^{*}$-algebras and $\varphi:A\rightarrow B$ be a linear
map. The map $\varphi$ is said to be positive if $\varphi(A_{+})\subset
B_{+}.$ The map $\varphi$ is said to be completely positive contractive,
abbreviated to c.p.c., if $\|\varphi\|\leq 1$ and
$\varphi\otimes\mathrm{id}:A\otimes M_{n}\rightarrow B\otimes M_{n}$ is
positive for all $n\in\mathbb{N}.$ A c.p.c. map $\varphi:A\to B$ is called
order zero, if for any $x,y\in A_{+},$ $xy=0$ implies $\varphi(x)\varphi(y)=0$
(see Definition 2.3 of [60]). If $ab=ba=0,$ we also write $a\perp b.$

In what follows, $\\{e_{i,j}\\}_{i,j=1}^{n}$ (or just $\\{e_{i,j}\\},$ if
there is no confusion) stands for a system of matrix units for $M_{n}$ and
$\iota\in C_{0}((0,1])$ is the identity function on $(0,1],$ i.e.,
$\iota(t)=t$ for all $t\in(0,1].$

######  Notation 2.3.

Throughout the paper, the set of all positive integers is denoted by
$\mathbb{N}.$ The set of all compact operators on a separable infinite-
dimensional Hilbert space is denoted by ${\cal K}.$

Let $A$ be a normed space and ${\cal F}\subset A$ be a subset. For any
$\epsilon>0$ and $a,b\in A,$ we write $a\approx_{\epsilon}b$ if
$\|a-b\|<\epsilon$. We write $a\in_{\varepsilon}{\cal F}$ if there is
$x\in{\cal F}$ such that $a\approx_{\varepsilon}x.$

######  Notation 2.4.

Let $\epsilon>0.$ Define a continuous function
$f_{\epsilon}:{{[0,+\infty)}}\rightarrow[0,1]$ by

$\displaystyle f_{\epsilon}(t)=\begin{cases}0&t\in{{[0,\epsilon/2]}},\\\
1&t\in[\epsilon,\infty),\\\
\mathrm{linear}&{t\in[\epsilon/2,\epsilon].}\end{cases}$

######  Definition 2.5.

Let $A$ be a $C^{*}$-algebra and $a,\,b\in(A\otimes{\cal K})_{+}.$ We write
$a\lesssim b$ if there is $x_{i}\in A\otimes{\cal K}$ for all $i\in\mathbb{N}$
such that $\lim_{i\rightarrow\infty}\|a-x_{i}^{*}bx_{i}\|=0$. We write $a\sim
b$ if $a\lesssim b$ and $b\lesssim a$ both hold. The Cuntz relation $\sim$ is
an equivalence relation. Set $\mathrm{Cu}(A)=(A\otimes{\cal K})_{+}/\sim.$ Let
$[a]$ denote the equivalence class of $a$. We write $[a]\leq[b]$ if $a\lesssim
b$.

######  Definition 2.6.

Let $A$ be a $\sigma$-unital $C^{*}$-algebra. A densely defined 2-quasi-trace
is a 2-quasi-trace defined on ${\rm Ped}(A)$ (see Definition II.1.1 of [5]).
Denote by ${\widetilde{QT}}(A)$ the set of densely defined quasitraces on
$A\otimes{\cal K}.$ Denote by ${\widetilde{T}}(A)$ the set of densely defined
traces on $A\otimes{\cal K}.$ In what follows we will identify $A$ with
$A\otimes e_{1,1},$ whenever it is convenient. Let
$\tau\in{\widetilde{QT}}(A).$ Note that $\tau(a)\not=\infty$ for any $a\in{\rm
Ped}(A)_{+}\setminus\\{0\\}.$

We endow ${\widetilde{QT}}(A)$ with the topology in which a net
${{\\{}}\tau_{i}{{\\}}}$ converges to $\tau$ if ${{\\{}}\tau_{i}(a){{\\}}}$
converges to $\tau(a)$ for all $a\in{\rm Ped}(A)$ (see also (4.1) on page 985
of [24]). A convex subset $S\subset{\widetilde{QT}}(A)\setminus\\{0\\}$ is a
basis for ${\widetilde{QT}}(A),$ if for any
$t\in{\widetilde{QT}}(A)\setminus\\{0\\},$ there exists a unique pair
$r\in\mathbb{R}_{+}$ and $s\in S$ such that $r\cdot s=t.$ Let $e\in{\rm
Ped}(A)_{+}\setminus\\{0\\}$ be a full element of $A.$ Denote by $QT(A)$ the
set of normalized 2-quasitraces of $A$ ($\|\tau\|=1$). Then
$S_{e}=\\{\tau\in{\widetilde{QT}}(A):\tau(e)=1\\}$ is a Choquet simplex and is
a basis for the cone ${\widetilde{QT}}(A)$ (see Proposition 3.4 of [55]).

Note that, for each $a\in({{A}}\otimes{\cal K})_{+}$ and $\varepsilon>0,$
$f_{\varepsilon}(a)\in{\rm Ped}(A\otimes{\cal K})_{+}.$ Define

$\displaystyle\widehat{[a]}(\tau):=d_{\tau}(a)=\lim_{\varepsilon\to
0}\tau(f_{\varepsilon}(a))\,\,\,{\rm
for\,\,\,all}\,\,\,\tau\in{\widetilde{QT}}(A).$ (e 2.1)

Except in subsection 2.2, we will assume that all 2-quasitraces of a separable
$C^{*}$-algebra $A$ in this paper are in fact traces for the convenience. This
is the case if $A$ is exact (by [32]).

######  Definition 2.7.

Let $A$ be a simple $C^{*}$-algebra with
$\widetilde{QT}(A)\setminus\\{0\\}\not=\emptyset.$ Then $A$ is said to have
(Blackadar’s) strict comparison, if, for any $a,b\in(A\otimes{\cal K})_{+},$
condition

$\displaystyle d_{\tau}(a)<d_{\tau}(b)\,\,\,{\rm
for\,\,\,all}\,\,\,\tau\in{\widetilde{QT}}(A)\setminus\\{0\\}$ (e 2.2)

implies that $a\lesssim b.$

######  Definition 2.8.

Let $A$ be a $C^{*}$-algebra with
${\widetilde{QT}}(A)\setminus\\{0\\}\not=\emptyset.$ Let
$S\subset{\widetilde{QT}}(A)$ be a convex subset. Set (if $0\not\in S,$ we
ignore the condition $f(0)=0$)

$\displaystyle\operatorname{Aff}_{+}(S)$ $\displaystyle=$
$\displaystyle\\{f:C(S,\mathbb{R})_{+}:f\,\,{{\rm affine}},f(s)>0\,\,{\rm
for}\,\,s\not=0,\,\,f(0)=0\\}\cup\\{0\\},$ (e 2.3) $\displaystyle{\rm
LAff}_{+}(S)$ $\displaystyle=$
$\displaystyle\\{f:S\to[0,\infty]:\exists\\{f_{n}\\},f_{n}\nearrow
f,\,\,f_{n}\in\operatorname{Aff}_{+}(S)\\}.$ (e 2.4)

For a simple $C^{*}$-algebra $A$ and each $a\in(A\otimes{\cal K})_{+},$ the
function $\hat{a}(\tau)=\tau(a)$ ($\tau\in S$) is in general in ${\rm
LAff}_{+}(S).$ If $a\in{\rm Ped}(A\otimes{\cal K})_{+},$ then
$\hat{a}\in\operatorname{Aff}_{+}(S).$ For $\widehat{[a]}(\tau)=d_{\tau}(a)$
defined above, we have $\widehat{[a]}\in{\rm LAff}_{+}({\widetilde{QT}}(A)).$

We write $\Gamma:\mathrm{Cu}(A)\to{\rm LAff}_{+}({\widetilde{QT}}(A))$ for the
canonical map defined by $\Gamma([a])(\tau)=\widehat{[a]}=d_{\tau}(a)$ for all
$\tau\in{\widetilde{QT}}(A).$

In the case that $A$ is algebraically simple (i.e., $A$ is a simple
$C^{*}$-algebra and $A={\rm Ped}(A)$), $\Gamma$ also induces a canonical map
$\Gamma_{1}:\mathrm{Cu}(A)\to{\rm LAff}_{+}(\overline{QT(A)}^{w}),$ where
$\overline{QT(A)}^{w}$ is the weak*-closure of $QT(A).$ Since, in this case,
$\mathbb{R}_{+}\cdot\overline{T(A)}^{w}={\widetilde{QT}}(A),$ the map $\Gamma$
is surjective if and only if $\Gamma_{1}$ is surjective. We would like to
point out that, in this case, $0\not\in\overline{QT(A)}^{w}$ (see Lemma 4.5 of
[22]).

In the case that $A$ is stably finite and simple, denote by
$\mathrm{Cu}(A)_{+}$ the set of purely non-compact elements (see Proposition
6.4 of [24]). Suppose that $\Gamma$ is surjective. Then
$\Gamma|_{\mathrm{Cu}(A)_{+}}$ is surjective as well (see Theorem 7.12 [26],
for example).

######  Definition 2.9.

Let $l^{\infty}(A)$ be the $C^{*}$-algebra of bounded sequences of $A.$ Recall
that $c_{0}(A):=\\{\\{a_{n}\\}\in
l^{\infty}(A):\lim_{n\to\infty}\|a_{n}\|=0\\}$ is a (closed two-sided) ideal
of $l^{\infty}(A).$ We view $A$ as a subalgebra of $l^{\infty}(A)$ via the
canonical map $\iota:a\mapsto\\{a,a,,...\\}$ for all $a\in A.$ In what
follows, we may identify $a$ with the constant sequence $\\{a,a,...,\\}$ in
$l^{\infty}(A)$ without further warning.

Put $A^{\prime}=\\{x=\\{x_{n}\\}\in
l^{\infty}(A):\lim_{n\to\infty}\|x_{n}a-ax_{n}\|=0\\}.$ It is called the
central sequence algebra of $A.$

######  Definition 2.10.

Let $A$ be a separable simple $C^{*}$-algebra with ${\rm Ped}(A)=A$ and
$QT(A)\not=\\{0\\}.$ Let $\tau\in{\widetilde{QT}}(A)\setminus\\{0\\}.$ Define,
for each $x\in A,$

$\displaystyle\|x\|_{{}_{2,\tau}}=\tau(x^{*}x)^{1/2}.$ (e 2.5)

Let $S\subset{\widetilde{QT}}(A)\setminus\\{0\\}$ be a compact subset. Define

$\displaystyle\|x\|_{{}_{2,S}}=\sup\\{\tau(x^{*}x)^{1/2}:\tau\in S\\}.$ (e
2.6)

Put $I_{S,\mathbb{N}}=\\{\\{x_{n}\\}\in
l^{\infty}(A):\lim_{n\to\infty}\|x\|_{{}_{2,S}}=0\\}.$

It is shown (Proposition 3.8 of [25]) that, with additional assumption that
$A$ has strict comparison,
$N_{cu}(A)=I_{{}_{\overline{QT(A)}^{w},\mathbb{N}}}.$

######  Definition 2.11.

Let $\varpi\in{\beta}(\mathbb{N})\setminus\mathbb{N}$ be a free ultrafilter.
Set

$\displaystyle c_{0,\varpi}=\\{\\{x_{n}\\}\in
l^{\infty}(A):\lim_{n\to\omega}\|x_{n}\|=0\\}.$ (e 2.7)

Let $S\subset{\widetilde{QT}}(A)$ be a compact subset. Define

$\displaystyle I_{{}_{S,\varpi}}=\\{\\{x_{n}\\}\in
l^{\infty}(A):\lim_{n\to\varpi}\|x_{n}\|_{{}_{2,S}}=0\\}.$ (e 2.8)

It is a (closed two-sided) ideal. In the case that $A={\rm Ped}(A),$ we
usually consider $I_{{}_{\overline{QT(A)}^{w},\varpi}}.$ If $A$ has continuous
scale, we consider $I_{{}_{QT(A),\varpi}}.$

Denote by $\Pi_{\varpi}:l^{\infty}(A)\to
l^{\infty}(A)/_{I_{\overline{QT(A)}^{w},\varpi}}$ the quotient map. If
$\tau\in{\widetilde{QT}}(A),$ for $x=\Pi_{\varpi}(\\{x_{n}\\}),$ define

$\displaystyle\tau_{\varpi}(x)=\lim_{n\to\varpi}\tau(x_{n}).$ (e 2.9)

We may view $\tau_{\varpi}$ as a trace on
$l^{\infty}(A)/_{I_{\overline{QT(A)}^{w},\varpi}}.$

If $S\subset QT(A)$ is a compact subset, denote by
$\pi_{S}:l^{\infty}(A)\to\l^{\infty}(A)/I_{{}_{S,\varpi}}$ and by
$\Phi_{S}:l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}\to
l^{\infty}/I_{{}_{S,\varpi}}$ the quotient maps, respectively. Note that $S$
could be a single point $\tau.$ For convenience, abusing the notation, we may
also write $A^{\prime}$ for $\pi_{S}(A^{\prime}).$

###  2.1 Extremal Boundaries

Throughout this paper, by a Choquet simplex, we mean a metrizable Choquet
simplex.

######  Definition 2.12.

Let $T$ be a Choquet simplex and $X=\partial_{e}(T)$ be the extremal boundary.
By the Choquet theorem, for each $t\in T,$ there is a unique Borel probability
measure $\mu_{t}$ on $X$ such that $t$ is the barycenter of $\mu_{t},$ i.e.,

$\displaystyle f(t)=\int_{X}f(x)d\mu_{t}\,\,\,{\rm
for\,\,\,all}\,\,\,f\in\operatorname{Aff}(T).$ (e 2.10)

Let $S\subset\partial_{e}(T)=X$ be a Borel subset. Denote by
$M_{S}=\\{\mu_{t}:t\in T\,\,\,{\rm and}\,\,\,\mu_{t}|_{X\setminus S}=0\\}.$ In
other words, $M_{S}$ is the set of those Borel probability measures on $X$
concentrated on $S.$ In what follows we may identify $t$ with $\mu_{t}$ and
$M_{S}$ with the subset of $T$ whose associated extremal boundary measure
concentrated on $S.$ Denote by ${\rm conv}(S)$ the convex hull of $S.$

We say $T$ satisfies condition (C), if $T$ has the following properties:

(1) $X=\cup_{n=1}^{\infty}X_{n},$ where each $X_{n}$ is a compact subset of
$X$ with finite covering dimension,

(2) for any $k,m\geq 1,$ $\overline{{\rm
conv}(\cup_{j=1}^{m}X_{j+k})}=M_{\cup_{j=1}^{m}X_{j+k}}$ and $\overline{{\rm
conv}(X\setminus\cup_{j=1}^{m}X_{j+k})}=M_{X\setminus\cup_{j=1}^{m}X_{j+k}}.$

In (1) above, we note that $\\{{\rm dim}X_{n}\\}$ may not be bounded. By an
elementary measure theory argument, if $S$ is a Borel subset of
$\partial_{e}(T),$ then one always has ${\rm conv}(S)\subset
M_{S}\subset\overline{{\rm conv}(S)}.$

If $S$ is a compact subset of $\partial_{e}(T),$ then, by Corollary 11.9 of
[31], $\overline{{\rm conv}(S)}$ is a closed face of $T$ and
$\partial_{e}(\overline{{\rm conv}(S)})=S.$ By the Choquet-Bisop-de Leeuw
theorem (cf. Theorem I.4.8 of [2]), for each $x\in\overline{{\rm conv}(S)},$
there exists a measure $m_{x}$ on $S$ such that, for any
$g\in\operatorname{Aff}(\overline{{\rm conv}(S)}),$ $g(x)=\int_{S}g(s)dm_{x}.$
Define ${\bar{m}}_{x}$ on $\partial_{e}(T)$ by ${\bar{m}}_{x}(B)=m_{x}(B\cap
S)$ for all Borel subsets $B$ of $\partial_{e}(T).$ Let $\mu_{x}$ be as
defined in (e 2.10). Then, by the Choquet theorem, for any
$f\in\operatorname{Aff}(T),$

$\displaystyle
f(x)=\int_{\partial_{e}(T)}f(t)d\mu_{x}=\int_{\partial_{e}(T)}f(t)d{\bar{m}}_{s}=\int_{S}f(s)dm_{s}.$
(e 2.11)

This implies that $\overline{{\rm conv}(S)}=M_{S}.$ In other words,
$\overline{{\rm conv}(\cup_{j=1}^{m}X_{j+k})}=M_{\cup_{j=1}^{m}X_{j+k}}$ is
automitic.

Note also that, if $S_{1},S_{2}\in X$ is a pair of disjoint subsets, then
$M_{S_{1}}\cap M_{S_{2}}=\emptyset.$ In particular, condition (2) implies that

$\displaystyle\overline{{\rm
conv}(\cup_{j=1}^{m}X_{j+k})}\,\cap\,\overline{{\rm
conv}(X\setminus\cup_{j=1}^{m}X_{j+k})}=\emptyset.$ (e 2.12)

It might be helpful to note that $k\geq 1$ in (2) above. One also sees
immediately from (2) that $X_{1}\cap X_{n}\not=\emptyset$ if $n\not=1.$ In
fact, (2) also implies that $X_{i}\cap X_{j}=\emptyset,$ if $i\not=j.$

Next we would like to provide some examples of Choquet simplexes $S$ which
satisfy condition (C). Three of them are not Bauer simplexes. Concrete
examples of $C^{*}$-algebras with non-Bauer simplexes as their tracial spaces
brought to our close attention are those tracial spaces from one-dimensional
NCCW complexes (see 3.8 of [28]). These can be “countablized” as follows.

######  Example 2.13.

(1) This example is the same as Example 3.3 of [11]. Denote by ${\bf c}$ the
$C^{*}$-algebra of convergent sequences. Let
$X=\\{0\\}\cup\\{1/k\\}_{k\in\mathbb{N}}\subset[0,1].$ We may view ${\bf
c}=C(X)$ as the algebra of continuous functions on $X.$ Let

$\displaystyle E_{4}=\\{x\in{\bf c}\otimes M_{2}:x(0)=\begin{pmatrix}a&0\\\
0&b\end{pmatrix},\,a,b\in\mathbb{C}\\}.$ (e 2.13)

This is one of the examples presented by L. G. Brown in [7] (p. 868) to
demonstrate different phenomena. Since $E_{4}$ is unital, $S=T(E_{4})$ is a
Choquet simplex. Let $\tau_{n}:E_{4}\to\mathbb{R}$ be defined by
$\tau_{n}(f)=t_{M_{2}}(f(1/n)),$ $n\in\mathbb{N}.$ Set $\tau^{+}(x)=a$ and
$\tau_{-}(x)=b$ (if $x(0)={\rm diag}(a,b)$). $C^{*}$-algebra $E_{4}$ is used
in Example 3.3 of [11] to show, among other things, that its tracial state
space is not a Bauer simplex. As in [11],
$\partial_{e}(S)=\\{\tau_{n}:n\in\mathbb{N}\\}\cup\\{\tau^{+},\tau^{-}\\}.$
Note (as in 3.3 of [11]) that $\\{\tau_{n}\\}_{n\in\mathbb{N}}$ converges to
$1/2(\tau^{+}+\tau^{-}).$ So $\partial_{e}(S)$ is not closed. Let
$X_{1}=\\{\tau^{+},\tau^{-}\\}$ and $X_{n+1}=\\{\tau_{n}\\},$
$n\in\mathbb{N}.$ Then it is easy to see that $S$ satisfies condition (C) but
is not a Bauer simplex.

(2) Define

$\displaystyle B=\\{x\in{\bf c}\otimes E_{4}:x(0)=\begin{pmatrix}a&0\\\
0&b\end{pmatrix},\,a,b\in\mathbb{C}\\}.$ (e 2.14)

Put $I=c_{0}\otimes E_{4}.$ We may view $I$ as an ideal of $B.$ Then
$B/I\cong\mathbb{C}\oplus\mathbb{C}.$ Note that $\partial_{e}(T(B))$ has only
countably many points but has infinitely many limit points and is not compact.
We may write
$\partial_{e}(T(B))=\\{0\\}\times\\{\tau^{+},\tau^{-}\\}\sqcup\cup_{n=1}^{\infty}\\{1/k\\}\times\partial_{e}(T(E_{4})).$
Note that, for each fixed $n,$ ${1/k}\times\tau^{+}\to\\{0\\}\times\tau^{+}$
and ${1/k}\times\tau^{-}\to\\{0\\}\times\tau^{-},$ as $k\to\infty,$
respectively. Define $X_{1}=X\times\\{\tau^{+},\tau^{-}\\}.$ Then $X_{1}$ is
compact and zero dimensional. Put $Y_{k,n}=\\{k\\}\times\\{\tau_{n}\\}.$ There
is a bijection $s:\mathbb{N}\to\\{(k,n):k,n\in\mathbb{N}\\}.$ Define
$X_{n+1}=Y_{s(n)},$ $n\in N.$ Then $X_{n+1}$ is compact and zero dimensional.

Denote by $E_{4,n}=\\{f\in E_{4}:f(1/j)=0\,\,\,{\rm
for\,\,\,all}\,\,\,j\not=n\\}.$ Then $E_{4,n}\cong M_{2}$ and is a direct
summand of $E_{4}.$ For each $i,j\in\mathbb{N},$

$\displaystyle B_{i,j}=\\{f\in B:f(1/k)=0,\,\,k\not=i,f(1/k)\in
E_{4,j}\\}\cong E_{4,j}\cong M_{2}.$ (e 2.15)

Then, for any $k,m\in\mathbb{N},$ $\bigoplus_{i=1}^{m}B_{s(i+k)}$ is
isomorphic to a finite direct sum of $E_{4,j}$ and

$\displaystyle B=B_{0}\bigoplus(\bigoplus_{i=1}^{m}B_{s(i+k)}),$ (e 2.16)

where $B_{0}\cong B.$ From this, it is easy to see that for
$k,m\in\mathbb{N},$

$\displaystyle\overline{{\rm
conv}(\cup_{i=1}^{m}X_{i+k})}=M_{\cup_{i=1}^{m}X_{i+k}}\,\,\,{\rm and}\,\,\,$
(e 2.17) $\displaystyle\overline{{\rm
conv}(X\setminus\cup_{i=1}^{m}X_{i+k})}=M_{X\setminus\cup_{i=1}^{m}X_{i+k}}$
(e 2.18)

In other words, $T(B)$ satisfies condition (C) but not a Bauer simplex.

(3) Let $\\{K_{n}\\}$ be a sequence of compact metric spaces with finite
covering dimension. Let $X$ be the one point compactification of disjoint
union of $\\{K_{n}\\}.$ Put $D_{1}=C(X).$ Then
$\partial_{e}(T(D_{1}))=\cup_{n=1}^{\infty}X_{n},$ where $X_{1}={\rm
the\,\,infinite\,\,point}=\\{\xi_{\infty}\\}$ and $X_{n+1}=K_{n},$
$n\in\mathbb{N}.$ It is easy to that $T(D_{1})$ satisfies condition (C). We
would like to mention that we may choose each $K_{n}$ the $n$-dimensional
cube, or the $n$-dimensional spheres, $n\in\mathbb{N},$ for example. Then $X$
is compact and has countable (but not necessarily finite) dimension.

(4) Let $X$ be as in (3). Define

$\displaystyle D_{2}=\\{f\in C(X,M_{2}):f(\xi_{\infty})=\begin{pmatrix}a&0\\\
0&b\end{pmatrix},\,a,b\in\mathbb{C}\\}.$ (e 2.19)

Then $D_{2}$ is a unital $C^{*}$-algebra, $T(D_{2})$ is a Choquet simplex and
$\partial_{e}(T(D_{2}))=\\{\tau^{+},\tau^{-}\\}\sqcup(\cup_{n=1}^{\infty}K_{n})$
which is not compact. Put $X_{1}=\\{\tau^{+},\tau^{-}\\}$ and $X_{n+1}=K_{n},$
$n\in\mathbb{N}.$ Then $\partial_{e}(T(D_{2}))=\cup_{n=1}^{\infty}X_{n},$
where each $X_{n}$ is compact and has finite covering dimension. As we
discussed in (1) above, $T(D_{2})$ satisfies condition (C) and is not a Bauer
simplex. Moreover $T(D_{2})$ is not finite dimensional, if $\\{{\rm
dim}X_{n}\\}$ is not bounded.

Similar to the construction of $B,$ one may also have a Choquet simplex $T$
which satisfies condition (C),
$\overline{\partial_{e}(T)}\setminus\partial_{e}(T)$ has infinitely many
points and $\partial_{e}(T)$ is not finite dimensional. If instead of
attaching two points to $\cup_{n=1}^{\infty}X_{n},$ we may attach a finite
dimensional compact metric space $Y$ to $\cup_{n=1}^{\infty}X_{n}.$ In that
way, some more interesting examples of Choquet simplex satisfying condition
(C) can be constructed.

(5) $C^{*}$-algebras $E_{4},$ $B$ and $D_{1}$ and $D_{2}$ are not simple. From
[3], any metrizable Choquet simplex $T$ can be realized as a tracial state
space of a unital simple AF-algebra. In fact $T$ can also be realized as a
tracial state space of a unital (or stably projectionless) simple
$C^{*}$-algebra $A$ which has arbitrary $K_{1}(A)$-group (and any compatible
$K_{0}(A)$) (see, for example, [28], [23] and [29]).

######  Lemma 2.14.

Let $T$ be a Choquet simplex such that (1) $\partial_{e}(T)=Y\cup Z$ and
$Y\cap Z=\emptyset,$ (2) $Y$ and $Z$ are Borel and, (3) $M_{Y}=\overline{{\rm
conv}(Y)}$ and $M_{Z}=\overline{{\rm conv}(Z)}.$

Suppose that $x_{n}={\alpha}_{n}y_{n}+(1-{\alpha}_{n})z_{n}\in T$ with
${\alpha}_{n}\in[0,1]$ and $y_{n}\in M_{Y}$ and $z_{n}\in M_{Z},$
$n\in\mathbb{N},$ and if $x_{n}\to{\alpha}y+(1-{\alpha})z\in T,$ where
${\alpha}\in[0,1],$ $y\in M_{Y}$ and $z\in M_{Z},$ then
${\alpha}_{n}\to{\alpha},$ $y_{n}\to y$ and $z_{n}\to z.$

###### Proof.

By the Choquet theorem, we identify $M_{Y}$ and $M_{Z}$ as convex subsets of
$T.$ By (1) and (3),
$T=\\{{\alpha}\mu_{y}+(1-{\alpha})\mu_{z}:0\leq{\alpha}\leq 1,\,\,\mu_{y}\in
M_{Y}\,\,\,{\rm and}\,\,\,\mu_{z}\in M_{Z}\\}.$ Let $t\in T$ be represented by
the Borel probability measure $\mu_{t}.$ Then
$\mu_{t}={\alpha}\mu_{y}+(1-{\alpha})\mu_{z}$ for some $\mu_{y}\in M_{Y},$
$\mu_{z}\in M_{Z}$ and ${\alpha}\in[0,1].$ In other words,
$\mu_{t}(Y)={\alpha}$ and $\mu_{t}(Z)=(1-{\alpha}).$ By the Choquet theorem,
this decomposition is unique. (It is also helpful to note that (1) and (3)
imply that $\overline{X}\cap\overline{Y}=\emptyset.$)

Now suppose that $x_{n}={\alpha}_{n}y_{n}+(1-{\alpha}_{n})z_{n}\in T$ with
${\alpha}_{n}\in[0,1]$ and $y_{n}\in M_{Y}=\overline{{\rm conv}(Y)}$ and
$z_{n}\in M_{Z}=\overline{{\rm conv}(Z)},$ $n\in\mathbb{N},$ and
$x_{n}\to{\alpha}y+(1-{\alpha})z\in T,$ where ${\alpha}\in[0,1],$
$y\in\overline{{\rm conv}(Y)}$ and $z\in\overline{{\rm conv}(Z)}.$ We will
show that ${\alpha}_{n}\to{\alpha},$ $y_{n}\to y$ and $z_{n}\to z.$

Otherwise there is a subsequence $\\{n_{k}\\}\subset\mathbb{N}$ such that

$\displaystyle{\rm dist}(z_{n_{k}},z)\geq\sigma$ (e 2.20)

for some $\sigma>0.$ We may assume that ${\alpha}_{n_{k}}\to{\alpha}_{0}$ for
some ${\alpha}_{0}\in[0,1].$ Since $T$ is compact, both $\overline{{\rm
conv}(Y)}$ and $\overline{{\rm conv}(Z)}$ are compact. Hence we may assume
that $z_{n_{k}}\to z_{0}\in\overline{{\rm conv}(Z)}$ and $y_{n_{k}}\to
y_{0}\in\overline{{\rm conv}(Z)}.$ Then

$\displaystyle{\alpha}_{n_{k}}y_{n_{k}}+(1-{\alpha}_{n_{k}})z_{n_{k}}\to{\alpha}_{0}y_{0}+(1-{\alpha})z_{0}.$
(e 2.21)

It follows that

$\displaystyle{\alpha}_{0}y_{0}+(1-{\alpha}_{0})z_{0}={\alpha}y+(1-{\alpha})z_{0}.$
(e 2.22)

Note that $y_{0},y\in M_{Y}$ and $z,z_{0}\in M_{Z}.$ By the conclusion of the
first paragraph of the proof, ${\alpha}_{0}={\alpha},$ $y_{0}=y$ and
$z=z_{0}.$ This leads a contradiction (to (e 2.20)) which proves the lemma. ∎

######  Corollary 2.15.

Let $T$ be a Choquet simplex such that (1) $\partial_{e}(T)=Y\cup Z$ and
$Y\cap Z=\emptyset,$ (2) $Y$ $Z$ are Borel and, (3) $M_{Y}=\overline{{\rm
conv}(Y)}$ and $M_{Z}=\overline{{\rm conv}(Z)}.$

Then there is $f\in\operatorname{Aff}(T)$ with $0\leq f\leq 1$ such that
$f|_{M_{Y}}=1$ and $f|_{M_{Z}}=0.$

###### Proof.

By (1) and (3), $\overline{Y}\cap\overline{Z}=\emptyset$ (in $T$). Define a
real affine function $f$ on $T$ such that $f|_{Y}=1$ and $f_{Z}=0$ and, for
each $t\in T,$

$\displaystyle f(t)=\int_{\partial_{e}(T)}f(x)d\mu_{t}.$ (e 2.23)

Clearly $f$ is affine (and continuous on $\partial_{e}(T)$). We also have
$f|_{M_{Y}}=1$ and $f|_{M_{Z}}=0.$ We will show that $f$ is continuous. Let
${\alpha}_{n}\in[0,1],$ $y_{n}\in M_{Y}$ and $z_{n}\in M_{Z}$ such that
${\alpha}_{n}y_{n}+(1-{\alpha}_{n})z_{n}\to{\alpha}y+(1-{\alpha})z,$ where
${\alpha}\in[0,1],$ $y\in M_{Y}$ and $z\in M_{Z}.$ Note that

$\displaystyle
f({\alpha}_{n}y_{n}+(1-{\alpha}_{n})z_{n})={\alpha}_{n}\,\,\,{\rm
and}\,\,\,f({\alpha}y+(1-{\alpha})z)={\alpha}.$ (e 2.24)

By Lemma 2.14, ${\alpha}_{n}\to{\alpha}.$ Therefore

$\displaystyle\lim_{n\to\infty}f({\alpha}_{n}y_{n}+(1-{\alpha}_{n})z_{n})=f{\alpha}y+(1-{\alpha})z).$
(e 2.25)

Thus $f$ is continuous. The lemma follows. ∎

######  Proposition 2.16.

Let $T$ be a Choquet simplex with condition (C) such that
$\partial_{e}(T)=\cup_{n=1}^{\infty}X_{n}$ which satisfies (1) and (2) in
2.12. Suppose that $x_{n}\in M_{\cup_{i=n}^{m(n)}X_{i}}$ (for some $m(n)\geq
n$) and $x_{n}\to x\in T$ (as $n\to\infty$). Then $x\in M_{X_{1}}.$

###### Proof.

Put $X=\partial_{e}(T)$ and fix $m\in\mathbb{N}\setminus\\{1\\}.$ Then
$x_{n}\in\overline{{\rm conv}(X\setminus\cup_{i=2}^{m}X_{i})}$ for all $n>m.$
It follows that $x\in\overline{{\rm conv}(X\setminus\cup_{i=2}^{m}X_{i})}.$
Put $Z_{m}=X\setminus\cup_{i=2}^{m}X_{i}$ and $Y_{m}=\cup_{i=2}^{m}X_{i}.$
Then, by 2.12, $x\in M_{Z_{m}}.$ It follows that the extremal boundary (Borel
probability) measure $\mu_{x}$ concentrates on $Z_{m}.$ In other words,
${\mu_{x}}|_{Y_{m}}=0.$ This holds for each $m.$ Therefore $x\in M_{X_{1}}.$ ∎

In the case that $A$ is not unital, we will consider the case that the cone
${\widetilde{T}}(A)$ has a basis $S$ which is a Choquet simplex satisfying
condition (C). We would like to point out that this condition does not depend
on the choice of basis $S.$ This is clarified by the following proposition.

######  Proposition 2.17.

Let $A$ be a separable simple $C^{*}$-algebra with
${\widetilde{T}}(A)\setminus\\{0\\}\not=\emptyset.$ Suppose that
$S\subset\widetilde{T}(A\otimes{\cal K})\setminus\\{0\\}$ is a compact Choquet
simplex which is also a basis for the cone $\widetilde{T}(A\otimes{\cal K}).$
Suppose that $e\in{\rm Ped}(A\otimes{\cal K})\setminus\\{0\\}$ with $0\leq
e\leq 1$ and $T_{e}=\\{\tau\in\widetilde{T}(A\otimes{\cal K}):\tau(e)=1\\}.$

(1) Then there is a homeomorphism (but not necessarily affine) $\gamma$ from
$S$ onto $T_{e}$ which maps $\partial_{e}(S)$ onto $\partial_{e}(T_{e}).$

(2) $S$ satisfies condition (C) if and only if $T_{e}$ satisfies condition
(C).

(3) If $S$ is a Bauer simplex, so is $T_{e}.$ Moreover, in this case, there is
an affine homeomorphism $\gamma:S\to T_{e}$ such that

$\displaystyle\gamma(\tau)(a)={\tau(a)\over{\tau(e)}}\,\,\,\text{for\,\,\,all}\,\,\,a\in{\rm
Ped}(A\otimes{\cal K})\,\,\,\text{and}\,\,\,\tau\in\partial_{e}(S).$ (e 2.26)

###### Proof.

Let $\gamma:S\to T_{e}$ be defined by $\gamma(s)(a)={s(a)\over{s(e)}}$ for all
$a\in{\rm Ped}(A\otimes{\cal K})$ and $s\in S.$ It is an injective continuous
map (recall that $A$ is simple, so $s(e)>0$). To see it is surjective, let
$\tau\in T_{e}.$ Then, since $S$ is a basis for the cone $\widetilde{T}(A),$
there is a unique $r\in\mathbb{R}_{+}\setminus\\{0\\}$ such that
$r\cdot\tau\in S.$ Thus

$\displaystyle\gamma(r\cdot\tau)(a)={r\cdot\tau(a)\over{r\cdot\tau(e)}}={\tau(a)\over{\tau(e)}}=\tau(a)$
(e 2.27)

for all $a\in{\rm Ped}(A\otimes{\cal K}).$ This shows that $\gamma$ is
surjective and hence a homeomorphism (between compact Hausdorff spaces).

To see $\gamma$ maps $\partial_{e}(S)$ to $\partial_{e}(T_{e}),$ let
$\tau\in\partial_{e}(S).$ Suppose that there are $\tau_{1},\tau_{2}\in T_{e}$
such that $\gamma(\tau)={\alpha}\tau_{1}+{\beta}\tau_{2},$ where
$0\leq{\alpha},\,{\beta}<1$ and ${\alpha}+{\beta}=1.$

Define $f_{S}\in{\rm Aff}(\widetilde{T}(A\otimes{\cal K}))$ by $f_{S}(r\cdot
s)=r$ for all $r\in\mathbb{R}$ and $s\in S.$ Recall that $S$ is a basis for
the cone $\widetilde{T}(A).$ In particular, if $s_{1},s_{2}\in S$ and
${\alpha}^{\prime}s_{1}+{\beta}^{\prime}s_{2}=s\in S$ for some
${\alpha}^{\prime},{\beta}^{\prime}>0,$ then
${\alpha}^{\prime}+{\beta}^{\prime}=1.$ To see this, let
$\delta={\alpha}^{\prime}+{\beta}^{\prime}>0.$ Then
$({\alpha}^{\prime}/\delta)s_{1}+({\beta}^{\prime}/\delta)s_{2}=s/\delta.$
Since $S$ is convex, $s_{3}=s/\delta\in S.$ Since $S$ is a basis, $\delta=1.$
From what we have just verified, $f_{S}$ is a well-defined real continuous
affine function, and $t\in S$ if and only if $f_{S}(t)=1.$ Moreover,
$f_{S}(\tau)>0,$ for any $\tau\in\widetilde{T}(A\otimes{\cal
K})\setminus\\{0\\}.$

Put $s(a)={\gamma(\tau)(a)\over{f_{S}(\gamma(\tau))}}$ for $a\in{\rm
Ped}(A\otimes{\cal K}).$ Then $s\in S.$ However, since
$\gamma(s)(e)=1=\gamma(\tau)(e),$

$\displaystyle\gamma(s)(a)={\gamma(\tau)(a)\over{f_{S}(\gamma(\tau))(\gamma(s)(e))}}={\gamma(\tau)(a)\over{f_{S}(\gamma(\tau))({f_{S}(\gamma(\tau))\over{\gamma(\tau)(e))}})}}=\gamma(\tau)(a)\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\rm Ped}(A\otimes{\cal K}).$ (e 2.28)

Since $\gamma$ is bijective, $s=\tau.$

On the other hand, define $t_{i}(a)=\tau_{i}(a)/s_{i}(f_{S})$ for all
$a\in{\rm Ped}(A\otimes{\cal K})$ and $i=1,2.$ Put
$\lambda={\alpha}f_{S}(\tau_{1})+{\beta}f_{S}(\tau).$ Then

$s=(({\alpha}\tau_{1}+{\beta}\tau_{2})/f_{S}({\alpha}\tau_{1}+{\beta}\tau_{2}))=({\alpha}\tau_{1}+{\beta}\tau_{2})/\lambda=({{\alpha}f_{S}(\tau_{1})\over{\lambda}})t_{1}+({{\beta}f_{S}(\tau_{2})\over{\lambda}})t_{2}.$

Since $s=\tau$ is assumed to be in $\partial_{e}(S)$ and
$({{\alpha}\tau_{1}(f_{S})\over{\lambda}})+({{\beta}\tau_{2}(f_{S})\over{\lambda}})=1,$
either

${{\alpha}f_{S}(\tau_{1})\over{\lambda}}=0,\,\,\,\text{or}\,\,\,{{\beta}\
f_{S}(\tau_{2})\over{\lambda}}=0,$

which forces either ${\alpha}=0$ or ${\beta}=0.$ This proves the fact that
$\gamma|_{\partial_{e}(S)}$ maps $\partial_{e}(S)$ to $\partial_{e}(T_{e}).$
Note that $\gamma^{-1}(t)(a)={t(a)\over{f_{S}(t)}}$ for all $t\in T_{e}$ and
$a\in{\rm Ped}(A\otimes{\cal K}).$ Similar argument shows that $\gamma^{-1}$
maps $\partial_{e}(T(pAp))$ to $\partial_{e}(T(A)).$ This implies that
$\gamma|_{\partial_{e}(S)}$ is a homeomorphism from $\partial_{e}(S)$ onto
$\partial_{e}(T_{e}).$ This proves the first part of the proposition.

For part (2), let $K$ be a Borel subset of $\partial_{e}(S).$ Note that, if
$\tau_{1},\tau_{2}\in K$ and ${\alpha}\in[0,1],$ then, with
$\tau={\alpha}\tau_{1}+(1-{\alpha})\tau_{2},$

$\displaystyle\gamma({\alpha}\tau_{1}+(1-{\alpha})\tau_{2})={{\alpha}\tau_{1}+(1-{\alpha})\tau_{2}\over{{\alpha}\tau_{1}(e)+(1-{\alpha})\tau_{2}(e)}}$
$\displaystyle=({{\alpha}\tau_{1}(e)\over{\tau(e)}})({\tau_{1}\over{\tau_{1}(e)}})+({(1-{\alpha})\tau_{2}(e)\over{\tau(e)}})({\tau_{2}\over{\tau_{2}(e)}})=({{\alpha}\tau_{1}(e)\over{\tau(e)}})\gamma(\tau_{1})+({(1-{\alpha})\tau_{2}(e)\over{\tau(e)}})\gamma(\tau_{2}).$

Since $\gamma(\tau)(e)=1,$ we have

$\displaystyle
1=\gamma({\alpha}\tau_{1}+(1-{\alpha})\tau_{2})(e)=({{\alpha}\tau_{1}(e)\over{\tau(e)}})+({(1-{\alpha})\tau_{2}(e)\over{\tau(e)}}).$
(e 2.29)

This shows that $\gamma$ maps ${\rm conv}(K)$ into ${\rm conv}(\gamma(K)).$ In
fact, since $\gamma(\tau)={\tau\over{\tau(e)}},$ from measure theory, we also
have $\gamma(M_{K})\subset M_{\gamma(K)}.$ Similarly, replacing $\hat{e}$ by
$f_{S}$ (in the proof of the first part), the same lines of the argument shows
that $\gamma^{-1}$ maps ${\rm conv}(\gamma(K))$ into ${\rm conv}(K).$ It
follows that $\gamma$ maps ${\rm conv}(K)$ onto ${\rm conv}(\gamma(K)).$ Since
$\gamma$ is a homeomorphism, $\gamma$ maps $\overline{{\rm conv}(K)}$ onto
$\overline{{\rm conv}(\gamma(K))}.$ Since ${\rm conv}(\gamma(K))\subset
M_{\gamma(K)}\subset\overline{{\rm conv}(\gamma(K))},$ if
$M_{K}=\overline{{\rm conv}(K)},$ then $M_{\gamma(K)}=\overline{{\rm
conv}(\gamma(K))}.$

Now if $S$ satisfies condition (C), we may write $S=\cup_{n=1}^{\infty}X_{n}$
which satisfies (1) and (2) in 2.12. Then
$T_{e}=\cup_{n=1}^{\infty}\gamma(X_{n})$ and each $\gamma(X_{n})$ is compact
and has finite covering dimension (as $X_{n}$ is compact and has finite
covering dimension). From what has been shown, if $k,m\in\mathbb{N},$

$\displaystyle\overline{{\rm
conv}(\cup_{j=1}^{m}\gamma(X_{j+k}))}=M_{\gamma(\cup_{j=1}^{m}X_{j+k})}\,\,\,{\rm
and}\,\,\,\overline{{\rm
conv}(X\setminus\cup_{j=1}^{m}\gamma(X_{j+k}))}=M_{T_{e}\setminus\cup_{j=1}^{m}\gamma(X_{j+k})}.$

Thus $T_{e}$ satisfies condition (C). The same argument shows that, if $T_{e}$
satisfies condition (C) so does $S.$

For (3), let us assume that $S$ is a Bauer simplex, i.e., $\partial_{e}(S)$ is
compact. By the first part of the proposition, so is $\partial_{e}(T_{e}).$

For each $\tau\in S,$ by the Choquet theorem, there is a unique probability
Borel measure $\mu_{\tau}$ on $\partial_{e}(S)$ such that

$f(\tau)=\int_{\partial_{e}(S)}fd\mu_{\tau}\,\,\,{\rm
for\,\,\,all}\,\,\,f\in\operatorname{Aff}(S).$

Define $\kappa(\mu_{\tau})(F)=\mu_{\tau}(\gamma^{-1}(F))$ for any Borel subset
$F\subset\partial_{e}(T_{e}).$ For each $g\in\operatorname{Aff}(T_{e}),$
$g|_{\partial_{e}(T_{e})}$ is continuous. For each $\tau\in S,$ define

$\displaystyle
g(\widetilde{\gamma}(\tau))=\int_{\partial_{e}(T_{e}))}gd\kappa(\mu_{\tau})\,\,\,{\rm
for\,\,\,all}\,\,\,g\in\operatorname{Aff}(T_{e}).$ (e 2.30)

Since $T_{e}$ is a Bauer simplex, the formula above gives the unique
continuous affine extension of $g|_{\partial_{e}(T_{e})}.$ Note that the map
$\gamma^{\sharp}:\operatorname{Aff}(T_{e})\to\operatorname{Aff}(S)$ by

$\displaystyle\gamma^{\sharp}(f)=\int_{\partial_{e}(S)}f\circ\gamma
d\mu_{\tau}\,\,\,{\rm for\,\,\,all}\,\,\,f\in\operatorname{Aff}(T_{e})$ (e
2.31)

is continuous in weak-* topology of $\operatorname{Aff}(T_{e}).$ It follows
that $\widetilde{\gamma}(\tau)\in T_{e}$ for each $\tau\in S$ and that the map
$\widetilde{\gamma}:S\to T_{e}$ defined by
$\tau\mapsto\widetilde{\gamma}(\tau)$ is continuous and affine. Since
$\widetilde{\gamma}|_{\partial_{e}(S)}=\gamma|_{\partial_{e}(S)}$ is a
homeomorphism, $\widetilde{\gamma}$ maps $S$ onto $T_{e},$ which is an affine
homeomorphism. ∎

######  Definition 2.18.

Let $A$ be a separable $C^{*}$-algebra. We say that extremal boundaries of the
tracial cone of $A$ satisfies condition (C) if
$\widetilde{T}(A)\setminus\\{0\\}\not=\emptyset$ and one of bases $S$ of
$\widetilde{T}(A)\setminus\\{0\\}$ satisfies condition (C).

###  2.2 Oscillation and orthogonal complements

######  Definition 2.19 (see Definition A.1 of [23]).

Let $A$ be a $C^{*}$-algebra Let $S\subset{\widetilde{QT}}\setminus\\{0\\}$ be
a compact subset. Define, for each $a\in{\rm Ped}(A\otimes{\cal K})_{+},$

$\displaystyle\omega(a)|_{S}$ $\displaystyle=$
$\displaystyle\inf\\{\sup\\{d_{\tau}(a)-\tau(c):\tau\in
S\\}:c\in\overline{a(A\otimes{\cal K})a},\,0\leq c\leq 1\\}.$ (e 2.32)

Recall (from Theorem 4.7 of [22]) that a $\sigma$-unital $C^{*}$-algebra $A$
is called compact, if ${\rm Ped}(A)=A.$ Then, by Lemma 4.5 of [22],
$0\not\in\overline{QT(A)}^{w}.$ If $A$ is compact, we may choose
$S=\overline{QT(A)}^{w}.$ In that case, we will omit $S$ in the notation. Note
that $\omega(a)=0$ if and only if $d_{\tau}(a)$ is continuous on $S.$

Let $A$ be a $\sigma$-unital simple $C^{*}$-algebra with
$\widetilde{QT}(A)\not=\\{0\\}.$ Let $e\in A$ be a strictly positive element.
If $A$ has continuous scale, then $\omega(e)=0$ and $QT(A)$ is compact (see,
for example, Proposition 5.4 of [22]). If $A$ also has strict comparison, then
$A$ has continuous scale if and only if $\omega(e)=0$ (see Proposition 5.4 and
Theorem 5.3 of [22]).

######  Definition 2.20 (Definition 4.7 of [26]).

Let $A$ be a $\sigma$-unital compact $C^{*}$-algebra with
$QT(A)\not=\emptyset.$ Let $a\in{\rm Ped}(A\otimes{\cal K})_{+}$ and let
$\Pi:l^{\infty}(A)\to l^{\infty}(A)/I_{{}_{\overline{QT(A)}^{w},\mathbb{N}}}$
be the quotient map. Define

$\displaystyle\Omega^{T}(a)=\inf\\{\|\Pi(\iota(a)-\\{b_{n}\\})\|:b_{n}\in\mathrm{Her}(a_{n})_{+},\|b_{n}\|\leq\|a\|,\,\,\lim_{n\to\infty}\omega(b_{n})|_{\overline{QT(A)}^{w}}=0\\}.$

One may call $\Omega^{T}(a)$ the tracial approximate oscillation of $a.$ If
$\Omega^{T}(a)=0,$ we say that the element $a$ has approximately tracial
oscillation zero. Note that $\Omega^{T}(a)=0$ if and only if there exists
$b_{n}\in\mathrm{Her}(a)_{+}$ with $\|b_{n}\|\leq\|a\|$ such that

$\displaystyle\lim_{n\to\infty}\|a-b_{n}\|_{{}_{2,\overline{QT(A)}^{w}}}=0\,\,\,{\rm
and}\,\,\,\lim_{n\to\infty}\omega(b_{n})=0$ (e 2.33)

(see Proposition 4.8 of [26]).

We say that $A$ has T-tracial approximate oscillation zero, if
$\Omega^{T}(a)=0$ for all $a\in{\rm Ped}(A\otimes{\cal K})_{+}$ (we still
assume that $A$ is compact). If we view
$\|\cdot\|_{{}_{2,\overline{QT(A)}^{w}}}$ as an $L^{2}$-norm, the condition
has an analogue to that “almost” continuous functions are $L^{2}$-norm dense.

If, in addition, $A$ has strict comparison, then we say $A$ has tracial
approximate oscillation zero. Tracial approximate oscillation can be defined
for $C^{*}$-algebras which do not have the property that ${\rm Ped}(A)=A.$
(see section 5 of [26]). We will use the following fact: If $A$ is a separable
simple $C^{*}$-algebra with $\widetilde{QT}(A)\not=\\{0\\}$ and the extremal
boundary $\partial_{e}(S)$ of a basis $S$ for the cone $\widetilde{QT}(A)$ has
only countably many points, then $A$ has T-tracial approximate oscillation
zero (Theorem 5.9 of [26]).

######  Definition 2.21.

Let $\varpi\in{\beta}(\mathbb{N})\setminus\mathbb{N}$ be a free ultrafilter.
Let $p\in l^{\infty}(A)/I_{{}_{\overline{QT(A)}^{w},\varpi}}$ be a projection.
An element $\\{e_{n}\\}\in l^{\infty}(A)_{+}^{\bf 1}$ is called a permanent
projection lifting of $p,$ if, for any sequence of integers $\\{m(n)\\},$
$\Pi_{\varpi}(\\{e_{n}^{1/m(n)}\\})=p.$

######  Proposition 2.22 (Proposition 6.2 of [26]).

Let $A$ be a separable compact $C^{*}$-algebra (see 2.19) with
$T(A)\not=\emptyset$ and let $\\{e_{n}\\}\in l^{\infty}(A)_{+}^{\bf 1}.$
Suppose that $\varpi\in{\beta}(\mathbb{N})\setminus\mathbb{N}$ is a free
ultrafilter.

(1) Suppose that $p=\Pi_{\varpi}(\\{e_{n}\\})$ is a projection. Then
$\\{f_{\delta}(e_{n})\\}$ is a permanent projection lifting of $p$ for any
$0<\delta<1/2$ and

$\displaystyle\lim_{n\to\varpi}\sup\\{\tau((e_{n}-f_{\delta}(e_{n})e_{n})):\tau\in\overline{QT(A)}^{w}\\}=0.\,\,\,$
(e 2.34)

(2) The following are equivalent:

(i) $\\{e_{n}\\}$ is a permanent projection lifting of
$p=\Pi_{\varpi}(\\{e_{n}\\}),$

(ii)
$\lim_{n\to\varpi}\sup\\{d_{\tau}(e_{n})-\tau(e_{n}^{2}):\tau\in\overline{QT(A)}^{w}\\}=0,$

(iii) for all $\delta\in(0,1/2),$

$\lim_{n\to\varpi}\sup\\{d_{\tau}(e_{n})-\tau(f_{\delta}(e_{n})):\tau\in\overline{QT(A)}^{w}\\}=0.$

###### Proof.

(1) Note that $\Pi(f_{\delta}(\\{e_{n}\\}))=f_{\delta}(\Pi(\\{e_{n}\\}))=p$
for any $0<\delta<1/2.$ Therefore $\Pi(f_{\delta/2}(\\{e_{n}\\})=p.$ Put
$b_{n}=f_{\delta}(e_{n}),$ $n\in\mathbb{N}.$ For any sequence of integers
$\\{m(n)\\},$

$\displaystyle b_{n}^{1/m(n)}\leq f_{\delta/2}(e_{n}),\,\,n\in\mathbb{N}.$ (e
2.35)

It follows that

$\displaystyle
p=\Pi_{\varpi}(\\{f_{\delta}(e_{n})\\})\leq\Pi_{\varpi}(\\{b_{n}^{1/m(n)}\\})\leq\Pi_{\varpi}(\\{f_{\delta/2}(e_{n})\\})=p.$
(e 2.36)

So $\\{b_{n}\\}$ is a permanent projection lifting of $p.$ Moreover
$(\\{e_{n}\\}-\\{f_{\delta}(e_{n})e_{n}\\})^{1/2}\in
I_{{}_{\overline{QT(A)}^{w},\varpi}}$ and (e 2.34) also holds. This proves
part (1) of the lemma.

(2) ((ii) $\Rightarrow$ (i)) If

$\lim_{n\to\varpi}\sup\\{d_{\tau}(e_{n})-\tau(e_{n}^{2}):\tau\in\overline{QT(A)}^{w}\\}=0,$

then $\\{e_{n}\\}-\\{e_{n}^{2}\\}\in I_{{}_{\overline{QT(A)}^{w},\varpi}}.$ So
$p=\Pi_{\varpi}(\\{e_{n}\\})$ is a projection. Morover, for any $\\{m(n)\\},$

$\displaystyle\sup\\{\tau(e_{n}^{1/m(n)})-\tau(e_{n}^{2}):\tau\in\overline{QT(A)}^{w}\\}\leq\sup\\{d_{\tau}(e_{n})-\tau(e_{n}^{2}):\tau\in\overline{QT(A)}^{w}\\}\to^{\varpi}0.$
(e 2.37)

It follows that $\\{e_{n}^{1/m(n)}-e_{n}^{2}\\}\in
I_{{}_{\overline{QT(A)}^{w},\varpi}}.$ Then
$\Pi_{\varpi}(\\{e^{1/m(n)}\\})=\Pi_{\varpi}(\\{e_{n}\\}).$ Therefore
$\\{e_{n}\\}$ is a permanent projection lifting of $p.$

((iii) $\Rightarrow$ (i)) Assume, for any $\delta\in(0,1/2),$

$\displaystyle\lim_{n\to\varpi}\sup\\{d_{\tau}(e_{n})-\tau(f_{\delta}(e_{n})):\tau\in\overline{QT(A)}^{w}\\}=0.$
(e 2.38)

Let $m(n)\in\mathbb{N}.$ There is, for each $n,$ an integer $k(n)\geq m(n)$
such that

$\displaystyle\|e_{n}^{1/k(n)}f_{\delta}(e_{n})-f_{\delta}(e_{n})\|<(1/2n)^{2}.$
(e 2.39)

Put $c_{n}=e_{n}^{1/k(n)}f_{\delta}(e_{n})-f_{\delta}(e_{n}).$ Then
$\Pi_{\varpi}(\\{c_{n}\\})=0.$ It then follows that

$\displaystyle\Pi_{\varpi}(\\{e_{n}^{1/k(n)}\\}-\\{f_{\delta}(e_{n})\\})=\Pi_{\varpi}(\\{e_{n}^{1/k(n)}-e_{n}^{1/k(n)}f_{\delta}(e_{n})\\}).$
(e 2.40)

However, for each $\tau\in\overline{T(A)}^{w},$

$\displaystyle\tau((e_{n}^{1/k(n)}-e_{n}^{1/k(n)}f_{\delta}(e_{n}))^{2})\leq\tau((e_{n}^{1/k(n)}-e_{n}^{1/k(n)}f_{\delta}(e_{n})))$
(e 2.41) $\displaystyle\leq
d_{\tau}(e_{n})-\tau(e^{1/k(n)}f_{\delta}(e_{n}))\leq
d_{\tau}(e_{n})-\tau(f_{\delta}(e_{n}))+(1/2n)^{2}.$ (e 2.42)

Hence

$\displaystyle\|e_{n}^{1/k(n)}-e_{n}^{1/k(n)}f_{\delta}(e_{n})\|^{2}_{{}_{2,\overline{T(A)}^{w}}}\leq\sup\\{d_{\tau}(e_{n})-\tau(f_{\delta}(e_{n})):\tau\in\overline{QT(A)}^{w}\\}+(1/2n)^{2}.$
(e 2.43)

It follows that
$\Pi_{\varpi}(\\{e_{n}^{1/k(n)}-e_{n}^{1/k(n)}f_{\delta}(e_{n})\\})=0.$ By (e
2.40), $\\{e_{n}^{1/k(n)}-f_{\delta}(e_{n})\\}\in
I_{{}_{\overline{QT(A)}^{w},\varpi}}.$ But
$\Pi_{\varpi}(\\{f_{\delta}(e_{n})\\})=\Pi_{\varpi}(\\{e_{n}\\}).$ It follows
that $\Pi_{\varpi}(\\{e_{n}^{1/k(n)}\\})=p.$ Recall that

$\displaystyle
p\leq\Pi_{\varpi}(\\{e_{n}^{1/m(n)}\\})\leq\Pi_{\varpi}(\\{e_{n}^{1/k(n)}\\})=p.$
(e 2.44)

We conclude that $p$ is a projection and $\\{e_{n}\\}$ is a permanent
projection lifting.

For the converse, we claim that if (i) holds then
$\lim_{n\to\varpi}\omega(e_{n})=0.$

Otherwise, there exists $\sigma>0$ satisfying the following: For any ${\cal
P}\in\varpi,$ there is an integer $n({\cal P})\in{\cal P}$ such that that
$\omega(e_{n({\cal P})})>\sigma.$ Fix any $\delta\in(0,1/4),$ by Proposition
4.6 of [26], for each of these $n({\cal P}),$ there is an integer $r(n({\cal
P}))$ such that

$\displaystyle\sup\\{\tau(e_{{}_{n({\cal P})}}^{1/r(n({\cal
P}))})-\tau(f_{\delta}(e_{{}_{n({\cal
P})}})):\tau\in\overline{T(A)}^{w}\\}>\omega(e_{{}_{n(\cal
P)}})-\sigma/4>\sigma/2.$ (e 2.45)

We now define, for each $k\in\mathbb{N},$ an integer $m(k)$ as follows: If
$k=n({\cal P})$ for some ${\cal P}\in\varpi,$ define

$\displaystyle m(k)=\min\\{r(n({\cal P})):k=n({\cal P})\\}.$ (e 2.46)

This is well defined. If $k\not=n({\cal P})$ for any ${\cal P}\in\varpi,$
define $m(k)=k.$

Then, for any ${\cal P}\in\varpi,$ if $k=n({\cal P})\in{\cal P},$ by (e 2.45),

$\displaystyle\sup\\{\tau(e_{k}^{1/m(k))})-\tau(f_{\delta}(e_{k})):\tau\in\overline{QT(A)}^{w}\\}>\sigma/2.$
(e 2.47)

In other words,

$\displaystyle\lim_{n\to\varpi}\|(e_{n}^{1/m(n)}-f_{\delta/2}(e_{n}))^{1/2}\|_{{}_{2,\overline{QT(A)}^{w}}}\not<\sigma/2.$
(e 2.48)

Therefore $\Pi(\\{e_{n}^{1/m(n)}\\})\not=\Pi(f_{\delta}(\\{e_{n}\\}))=p.$ A
contradiction. Hence $\lim_{n\to\varpi}\omega(e_{n})=0.$

For (i) $\Rightarrow$ (ii), and (i) $\Rightarrow$ (iii), we first note that
there exists a sequence $\delta_{n}\in(0,1/2)$ such that
$\lim_{n\to\infty}\delta_{n}=0$ and

$\sup\\{d_{\tau}(e_{n})-\tau(f_{\delta_{n}}(e_{n})):\tau\in\overline{QT(A)}^{w}\\}<\omega(e_{n})+1/n,\,\,n\in\mathbb{N}.$

As in the proof of (iii) $\Rightarrow$ (i), we may choose integers $m(n)$ such
that

$\displaystyle\|e_{n}^{1/m(n)}f_{\delta_{n}}(e_{n})-f_{\delta_{n}}(e_{n})\|<1/n,\,\,n\in\mathbb{N}\,\,\,{\rm
and}\,\,\,$ (e 2.49)
$\displaystyle\sup\\{d_{\tau}(e_{n})-\tau(e_{n}^{1/m(n)}):\tau\in\overline{QT(A)}^{w}\\}<\omega(e_{n})+1/n,\,\,n\in\mathbb{N}.$
(e 2.50)

Put $d_{n}=e_{n}^{1/m(n)}f_{\delta_{n}}(e_{n})-f_{\delta_{n}}(e_{n}),$
$n\in\mathbb{N}.$ Then $\\{d_{n}\\}\in c_{0}(A).$ Note that
($e_{n}^{2}e_{n}^{1/m(n)}=e_{n}^{1/m(n)}e_{n}^{2}$)

$\displaystyle\sup\\{d_{\tau}(e_{n})-\tau(e_{n}^{2}):\tau\in\overline{QT(A)}^{w}\\}$
$\displaystyle\leq\sup\\{d_{\tau}(e_{n})-\tau(e_{n}^{1/m(n)}):\tau\in\overline{QT(A)}^{w}\\}+\sup\\{\tau(e_{n}^{1/m(n)}-e_{n}^{2}):\tau\in\overline{QT(A)}^{w}\\}$
$\displaystyle<\omega(e_{n})+1/n+\|e_{n}^{1/m(n)}-e_{n}^{2}\|_{2,\overline{QT(A)}^{w}}.$
(e 2.51)

By the claim just proved, if (i) holds, then (ii) follows.

Similarly, we have

$\displaystyle\sup\\{d_{\tau}(e_{n})-\tau(f_{\delta}(e_{n})):\tau\in\overline{QT(A)}^{w}\\}$
$\displaystyle\leq\sup\\{|d_{\tau}(e_{n})-\tau(f_{\delta_{n}}(e_{n}))|:\tau\in\overline{QT(A)}^{w}\\}+\sup\\{|\tau(f_{\delta_{n}}(e_{n}))-\tau(f_{\delta}(e_{n}))|:\tau\in\overline{QT(A)}^{w}\\}$
$\displaystyle<\omega(e_{n})+1/n+\|f_{\delta_{n}}(e_{n})-f_{\delta}(e_{n})\|_{{}_{2,\overline{QT(A)}^{w}}}.$
(e 2.52)

We note that

$\displaystyle e_{n}^{1/m(n)}\geq
e_{n}^{1/m(n)}f_{\delta_{n}}(e_{n})=f_{\delta_{n}}(e_{n})+d_{n},\,\,n\in\mathbb{N}.$
(e 2.53)

Since $d_{n}\in c_{0}(A),$

$\Pi_{\varpi}(\\{f_{\delta_{n}}(e_{n})\\})=\Pi_{\varpi}(\\{f_{\delta_{n}}(e_{n})+d_{n}\\})=\Pi_{\varpi}(e^{1/m(n)}f_{\delta_{n}}(e_{n})\\})\leq\Pi_{\varpi}(\\{e_{n}^{1/m(n)}\\})=p.$

Since $\lim_{n\to\infty}\delta_{n}=0,$ we also have (for large $n$)

$\displaystyle
p=\Pi_{\varpi}(\\{f_{\delta}(e_{n})\\})\leq\Pi_{\varpi}(\\{f_{\delta_{n}}(e_{n})\\})\leq
p.$ (e 2.54)

Consequently

$\\{f_{\delta_{n}}(e_{n})-f_{\delta}(e_{n})\\}\in
I_{{}_{\overline{T(A)}^{w},\varpi}}.$

Then, by (e 2.52) and the claim that $\lim_{n\to\infty}\omega(e_{n})=0,$ we
have

$\displaystyle\lim_{n\to\varpi}\sup\\{d_{\tau}(e_{n})-\tau(f_{\delta}(e_{n})):\tau\in\overline{QT(A)}^{w}\\}=0.$
(e 2.55)

This implies (iii) holds. The proposition follows. ∎

For the rest of the paper, we will assume that $C^{*}$-algebra $A$ has the
property that all 2-quasitraces are traces.

Proposition 2.24 and Corollary 2.25 are not directly used in the proof of the
next two sections. However, some ideas behind the proof of section 4 are
inspired by (a somewhat more complicated version of) these easy facts. We
believe that it is appropriate to share these with the reader.

######  Remark 2.23.

Let $A$ be a separable $C^{*}$-algebra and $a\in A_{+}^{\bf 1}.$ Put
$C=\mathrm{Her}(a)$ and $C^{\perp}=\\{b\in A:ba=ab=0\\}.$ A natural but naive
question is how large is $C+C^{\perp}?$ The answer is, of course, very
disappointing, as $C^{\perp}$ may well be zero. Let $p_{a}$ be the open
projection associated with $a.$ It is well known that $p$ may well be dense in
$A$ (even in the commutative case).

We would like to offer Proposition 2.24 and Corollary 2.25 to rescue us from
the total disaster as we will see $C+C^{\perp}$ could be approximately as
large as possible (at least tracially).

######  Proposition 2.24.

Let $A$ be a compact $C^{*}$-algebra with
$\widetilde{T}(A)\setminus\\{0\\}\not=\emptyset$ and let $e_{B}\in
A_{+}\setminus\\{0\\}.$ Suppose that $0\leq a\leq 1$ is in $B={\rm
Her}(e_{B}).$ Then, for any $\varepsilon>0,$ there exists $\delta_{0}>0$ such
that, for all $0<\delta<\delta_{0},$

$\displaystyle d_{\tau}(f_{\delta}(a))+d_{\tau}(e_{C})$ $\displaystyle>$
$\displaystyle
d_{\tau}(e_{B})-\liminf_{\eta\to\delta/2}\\{d_{\tau}(f_{\eta}(a))-d_{\tau}(f_{\delta}(a)):0<\eta<\delta/2\\}$
(e 2.56) $\displaystyle>$ $\displaystyle
d_{\tau}(e_{B})-\omega(a)-\varepsilon\hskip
21.68121pt\,\,\,\text{for\,\,\,all}\,\,\,\tau\in\overline{T(A)}^{w},$ (e 2.57)

where $e_{C}$ is a strictly positive element of $C={\rm
Her}(f_{\delta}(a))^{\perp}\cap B.$ Moreover,

$\displaystyle\omega(f_{\delta}(a)+e_{B})<\omega(e_{C})+\omega(a)+\varepsilon.$
(e 2.58)

###### Proof.

Let $\varepsilon>0.$ Choose $\delta_{0}>0$ such that, for all $0<r\leq
2\delta_{0},$

$\displaystyle\sup\\{d_{\tau}(a)-\tau(f_{r}(a)):\tau\in\overline{T(A)}^{w}\\}<\omega(a)+\varepsilon/4.$
(e 2.59)

Set $0<\delta<\delta_{0}.$ Choose any $0<\eta<\delta/2.$ Define, for
$m,n\in\mathbb{N},$

$d_{m,n}:=(1-f_{\eta}(a)^{1/n})e_{B}^{1/m}(1-f_{\eta}(a)^{1/n}).$

Then $0\leq d_{m,n}\leq 1$ and

$\displaystyle d_{m,n}f_{\delta}(a)=f_{\delta}(a)d_{m,n}=0.$ (e 2.60)

In other words, $d_{m,n}\in C:={\rm Her}(f_{\delta}(a))^{\perp}\cap B.$ Let
$e_{C}\in C$ be a strictly positive element. We compute that, for all
$\tau\in\overline{T(A)}^{w},$

$\displaystyle
d_{\tau}(e_{C})\geq\tau(d_{n,m})=\tau(e_{B}^{1/m})-2\tau(e_{B}^{1/m}f_{\eta}(a)^{1/n})+\tau(f_{\eta}(a)^{1/n}e_{B}^{1/m}f_{\eta}(a)^{1/n}).$
(e 2.61)

Let $m\to\infty,$ for all $\tau\in\overline{T(A)}^{w},$

$\displaystyle d_{\tau}(e_{C})\geq
d_{\tau}(e_{B})-2\tau(f_{\eta}(a)^{1/n})+\tau(f_{\eta}(a)^{2/n}).$ (e 2.62)

Then, by letting $n\to\infty,$ we obtain $d_{\tau}(e_{C})\geq
d_{\tau}(e_{B})-d_{\tau}(f_{\eta}(a)).$ Therefore

$\displaystyle d_{\tau}(f_{\delta}(a))+d_{\tau}(e_{C})\geq
d_{\tau}(e_{B})-(d_{\tau}(f_{\eta}(a))-d_{\tau}(f_{\delta}(a))\,\,\,{\rm
for\,\,\,all}\,\,\,\tau\in\overline{T(A)}^{w}.$ (e 2.63)

Letting $\eta\to\delta/2,$ we obtain, for all $\tau\in\overline{T(A)}^{w},$

$\displaystyle d_{\tau}(f_{\delta}(a))+d_{\tau}(e_{C})\geq
d_{\tau}(e_{B})-\liminf_{\eta\to\delta}\\{d_{\tau}(f_{\eta}(a))-d_{\tau}(f_{\delta}(a)):0<\eta<\delta/2\\}.$
(e 2.64)

However, for any $\tau\in\overline{T(A)}^{w},$

$\displaystyle\liminf_{\eta\to\delta}\\{d_{\tau}(f_{\eta}(a))-d_{\tau}(f_{\delta}(a)):0<\eta<\delta/2\\}\leq
d_{\tau}(a)-\tau(f_{\delta_{0}}(a))\leq\omega(a)+\varepsilon.$ (e 2.65)

We then have, for all $\tau\in\overline{T(A)}^{w},$

$\displaystyle d_{\tau}(f_{\delta}(a))+d_{\tau}(e_{C})$ $\displaystyle\geq$
$\displaystyle
d_{\tau}(e_{B})-\liminf_{\eta\to\delta}\\{d_{\tau}(f_{\eta}(a))-d_{\tau}(f_{\delta}(a)):0<\eta<\delta/2\\}$
(e 2.66) $\displaystyle>$ $\displaystyle
d_{\tau}(e_{B})-\omega(a)+\varepsilon.$ (e 2.67)

For the last part of the statement, since $d_{mn}\in C,$ we may choose $e_{C}$
so that $d_{m,n}\leq e_{C}.$ Then, by I.1.11 of [5],

$\displaystyle(1/2)e_{B}\leq d_{m,n}+f_{\eta}(a)^{1/n}\leq
f_{\delta}(a)+e_{C}+(f_{\eta}(a)^{1/n}-f_{\delta}(a)).$ (e 2.68)

It follows that $e_{B}\sim
f_{\delta}(a))+e_{C}+(f_{\eta}(a)^{1/n}-f_{\delta}(a)).$ Therefore (by (1) of
Proposition 4.4 of [26] for the first inequality below),

$\displaystyle\omega(e_{B})$ $\displaystyle=$
$\displaystyle\omega(f_{\delta}(a)+e_{C}+(f_{\eta}(a)^{1/n}-f_{\delta}(a)))$
(e 2.69) $\displaystyle\geq$
$\displaystyle\omega(f_{\delta}(a)+e_{C})-\sup\\{d_{\tau}(f_{\eta}(a)^{1/n}-f_{\delta}(a)):\tau\in\overline{T(A)}^{w}\\}$
(e 2.70) $\displaystyle{\stackrel{{\scriptstyle\eqref{LOs-n1-nn1}}}{{>}}}$
$\displaystyle\omega(f_{\delta}(a)+e_{C})-\omega(a)-\varepsilon/4.$ (e 2.71)

Hence

$\displaystyle\omega(f_{\delta}(a))+e_{C})<\omega(e_{B})+\omega(a)+\varepsilon.$

∎

######  Corollary 2.25.

Let $A$ be a compact $C^{*}$-algebra with
$\tilde{T}(A)\setminus\\{0\\}\not=\emptyset$ and let $e_{B}\in
A_{+}\setminus\\{0\\}.$ Suppose that $0\leq a\leq 1$ is in $B:={\rm
Her}(e_{B})$ with $\omega(a)=0.$ Then, for any $\varepsilon>0,$ there exists
$\delta_{0}>0$ such that, for all $0<\delta<\delta_{0},$

$\displaystyle
d_{\tau}(f_{\delta}(a))+d_{\tau}(e_{C})>d_{\tau}(e_{B})-\varepsilon\,\,\,\text{for\,\,\,all}\,\,\,\tau\in\overline{T(A)}^{w},$
(e 2.72)

where $e_{C}$ is a strictly positive element of $C:={\rm
Her}(f_{\delta}(a))^{\perp}\cap B.$ Moreover,

$\displaystyle\omega(f_{\delta}(a)+e_{C})<\omega(e_{B})+\varepsilon.$

##  3 Lifting

Throughout this section $\varpi\in{\beta}(\mathbb{N})\setminus\mathbb{N}$ is a
fixed free ultrafilter.

The most of the content of this section is within the same lines as that of
[43] which are also reformulated in [35] and [58]. But by the end of this
section, we add some flavor of tracial approximate oscillation to these
results.

The following is a version of a result of Y. Sato (see Lemma 2.1 of [51]). The
unital version (see Theorem 3.3 of [35]) of the next lemma can also be found
in [35] (for the second statement, see also the end of Remark 3.11 of [36]).

######  Theorem 3.1.

Let $A$ be a separable $C^{*}$-algebra, $\tau$ a faithful tracial state, $N$
the weak closure of $A$ under the GNS representation of $A$ with respect the
state $\tau.$ Then there are natural surjective homomorphisms

$\displaystyle l^{\infty}(A)/c_{\varpi}(A)\to
l^{\infty}(N)/I_{\tau,\varpi}(N)\,\,\,\text{and}\,\,\,l^{\infty}(A)/c_{\varpi}(A)\cap
A^{\prime}\to(l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N))\cap N^{\prime}.$ (e 3.1)

Moreover,

$\displaystyle l^{\infty}(A)/I_{{}_{\tau,\varpi}}\cong
l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N)\,\,\,\text{and}\,\,\,(l^{\infty}(A)/I_{{}_{\tau,\varpi}})\cap
A^{\prime}\cong(l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N))\cap N^{\prime}.$ (e
3.2)

###### Proof.

The unital case of the theorem is known (see Theorem 3.3 of [35]). The proof
of the non-unital case is the same. However, since we will study non-unital
$C^{*}$-algebras, we would like to repeat the proof of the first part of the
statement only for the convenience of the reader.

Let $\pi_{A}$ and $\pi_{N}$ denote the quotient maps $l^{\infty}(A)\to
l^{\infty}(A)/c_{\varpi}(A)$ and $l^{\infty}(N)\to
l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N),$ respectively. Denote the canonical map
$l^{\infty}(A)/c_{\varpi}(A)\to l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N)$ by
$\Phi,$ and let $\widetilde{\Phi}:l^{\infty}(A)\to
l^{\infty}(N)/c_{\varpi}(N)$ denote the map $\Phi\circ\pi_{A}.$

We will only repeat the proof for non-unital case which is the same as the
proof of unital case for the map $\Phi:l^{\infty}(A)/c_{\varpi}(A)\to
l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N).$

Let $x=\pi_{A}(\\{x_{n}\\})\in l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N).$ Then,
by Kaplanski’s density theorem (we wrote this on a Thanksgivings Day which is
not a Sunday–see 2.3.4 of [45]), there exists $a_{k}\in A$ with
$\|a_{k}\|\leq\|x_{k}\|$ such that $\|a_{k}-x_{k}\|_{2,\tau}<1/k.$ It follows
that $a:=\\{a_{k}\\}\in l^{\infty}(A)$ and $\widetilde{\Phi}(a)=x$ ($A$ has
unit or not). This shows that the proof for Theorem 3.3 of [35] works for non-
unital case. We are allowed not to repeat the same proof for the second
surjectivity.

∎

The following lemma is stated exactly the same as Lemma 2.9 of [58] for the
unital case.

######  Lemma 3.2 (Lemma 3.3 of [43] and Lemma 2.9 of [58]).

Let $A$ be a separable non-elementary simple nuclear $C^{*}$-algebra with
continuous scale and $\tau$ be an extremal tracial state. Then, for any $k\geq
2,$ there exists a unital homomorphism
$\varphi:M_{k}\to(l^{\infty}(A)/I_{{}_{\tau,\varpi}})\cap A^{\prime}$ and an
order zero c.p.c. map $\psi:M_{k}\to(l^{\infty}(A)/c_{\varpi}(A))\cap
A^{\prime}$ such that $\pi\circ\psi=\varphi,$ where
$\pi:(l^{\infty}(A)/c_{\varpi}(A))\cap
A^{\prime}\to(l^{\infty}(A)/I_{{}_{\tau,\varpi}})\cap A^{\prime}.$

###### Proof.

The proof is the same as that for the unital case. To be complete, let us
repeat some of it. Let $\rho_{\tau}$ denote the GNS-representation associated
with $\tau.$ Since $\tau$ is an extremal point of $T(A),$
$N:=\rho_{\tau}(A)^{\prime\prime}$ is a factor. This is standard. Otherwise,
let $p\in N^{\prime}$ be a non-zero projection such that $p\not=1.$ Define two
non-zero traces $\tau_{1},\tau_{2}$ as follows:

$\displaystyle\tau_{1}(a):=\tau(p\pi_{\tau}(a))\,\,\,{\rm
and}\,\,\,\tau_{2}:=\tau((1-p)\pi_{\tau}(a))\,\,\,{\rm for\,\,\,all}\,\,\,a\in
A.$ (e 3.3)

Note that $0<\|\tau_{i}\|\leq 1,$ $i=1,2.$ Put $t_{i}=\tau_{i}/\|\tau_{i}\|,$
$i=1,2.$ Then $t_{i}\in T(A),$ $i=1,2$ and

$\displaystyle\tau=\|\tau_{1}\|t_{1}+\|\tau_{2}\|t_{2}.$ (e 3.4)

This contradicts the fact that $\tau$ is an extremal point of $T(A).$

Since $A$ is nuclear, $N$ is an injective ${\rm II}_{1}$-factor and hence
McDuff (see Theorem 5.1 of [13]). It follows that there is a unital embedding
$\varphi:M_{k}\to(l^{\infty}(N)/I_{{}_{\varpi,\tau}}(N))\cap
N^{\prime}\cong(l^{\infty}(A)/I_{{}_{\tau,\varpi}})\cap A^{\prime}.$ By
Theorem 3.1, there is an order zero c.p.c. map
$\psi:M_{k}\to(l^{\infty}(A)/c_{\varpi}(A))\cap A^{\prime}$ such that
$\varphi=\pi\circ\psi$ as desired. ∎

######  Lemma 3.3.

Let $A$ be a separable simple $C^{*}$-algebra with ${\rm Ped}(A)=A,$ $S\subset
T(A)$ a compact subset and $p\in l^{\infty}(A)/I_{{}_{S,\varpi}}$ a
projection. Suppose that $A$ has T-tracial approximate oscillation zero (and
all 2-quasitraces of $A$ are traces). Then there is a permanent projection
lifting $\\{e_{n}\\}\in l^{\infty}(A)$ of projection
$q=\Pi_{\varpi}(\\{e_{n}\\})$ such that $\pi_{S}(\\{e_{n}\\})=p$ and
$\Phi_{S}(q)=p,$ where $\pi_{S}:l^{\infty}(A)\to
l^{\infty}(A)/I_{{}_{S,\varpi}}$ and
$\Phi_{S}:l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}\to
l^{\infty}(A)/I_{{}_{S,\varpi}}$ are quotient maps, respectively.

Moreover, if $\\{a_{n}\\}\in l^{\infty}(A)_{+}^{\bf 1}$ is a given lifting of
$p,$ we may choose $e_{n}\in\mathrm{Her}(a_{n})_{+}^{\bf 1},$
$n\in\mathbb{N}.$

###### Proof.

Let $\\{a_{n}\\}\in l^{\infty}(A)_{+}^{\bf 1}$ be such that
$\pi_{S}(\\{a_{n}\\})=p.$ Since $A$ has $T$-tracial approximate oscillation
zero, for each $n,$ there exists
$b_{n}\in\mathrm{Her}(f_{1/4}(a_{n}))_{+}^{\bf 1}$ such that

$\displaystyle\|f_{1/4}(a_{n})-b_{n}\|_{{}_{2,\overline{T(A)}^{w}}}<1/(n+1)\,\,\,{\rm
and}\,\,\,\omega(b_{n})<1/(n+1)^{2},\,\,n\in\mathbb{N}.$ (e 3.5)

It follows that

$\displaystyle p=\pi_{S}(f_{1/4}(a_{n}))=\pi_{S}(\\{b_{n}\\})\,\,\,{\rm
and}\,\,\,\Pi_{\varpi}(\\{f_{1/4}(a_{n})\\})=\Pi_{\varpi}(\\{b_{n}\\}).$ (e
3.6)

Choose $m(n)\in\mathbb{N}$ such that (see 2.19)

$\displaystyle d_{\tau}(b_{n})-\tau(b_{n}^{2/m(n)})<1/n^{2}\,\,\,{\rm
for\,\,\,all}\,\,\,\tau\in\overline{T(A)}^{w}.$ (e 3.7)

Since $f_{1/8}(a_{n})b_{n}^{1/m(n)}=b_{n}^{1/m(n)}$ for all $n\in\mathbb{N},$
we have

$\displaystyle
p=\pi_{S}(\\{b_{n}\\})\leq\pi_{S}(\\{b_{n}^{1/m(n)}\\})\leq\pi_{S}(\\{f_{1/8}(a_{n})\\})=f_{1/8}(\pi_{S}(\\{a_{n}\\}))=p.$
(e 3.8)

Put $e_{n}=\\{b_{n}^{1/m(n)}\\},$ $n\in\mathbb{N}.$ Then
$p=\pi_{S}(\\{e_{n}\\}).$ Note that (e 3.7) also implies that

$\displaystyle\|b_{n}^{1/m(n)}-(b_{n}^{1/m(n)})^{2}\|_{{}_{2,\overline{T(A)}^{w}}}<1/n,\,\,n\in\mathbb{N}.$
(e 3.9)

It follows that

$\displaystyle\Pi_{\varpi}(\\{e_{n}\\})^{2}=\Pi_{\varpi}(\\{e_{n}\\}).$ (e
3.10)

Then $q=\Pi_{\varpi}(\\{e_{n}\\})$ is a projection. By (e 3.7) and (2) of
Proposition 2.22, $\\{e_{n}\\}$ is a permanent projection lifting of $q.$
Moreover $\Phi_{S}(q)=p.$

For the last statement, just recall that
$e_{n}\in\mathrm{Her}(f_{1/4}(a_{n}))_{+}^{\bf 1},$ $n\in\mathbb{N}.$ ∎

It is proved in [26] (see Theorem 6.4 of [26]) that, when $A={\rm Ped}(A)$ and
$A$ has T-tracial approximate oscillation zero,
$l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\mathbb{N}}}$ has real rank
zero111In [26], it is actually shown that
$l^{\infty}(A)/I_{{}_{\overline{QT(A)}^{w}}}$ has real rank zero. Since
$l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w}}}$ is a quotient of
$l^{\infty}(A)/I_{{}_{\overline{QT(A)}^{w}}},$ it also has real rank zero.. It
then follows that $I_{{}_{S,\varpi}}/I_{{}_{\overline{T(A)}^{w},\varpi}}$ and
$l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}$ have real rank zero. The
short exact sequence considered in the next lemma is

$\displaystyle{\small{0\to
I_{{}_{S,\varpi}}/I_{{}_{\overline{T(A)}^{w},\varpi}}\to
l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}\to
l^{\infty}(A)/I_{{}_{S,\varpi}}\to 0.}}$ (e 3.11)

Thus the next lemma is a non-$\sigma$-unital version of the Elliott Lifting
Lemma (see Corollary 3.3 of [16]).

######  Lemma 3.4.

Let $A$ be a separable simple $C^{*}$-algebra with ${\rm Ped}(A)=A$ which has
$T$-tracial approximate oscillation zero, and let $S\subset T(A)$ be a compact
subset. Suppose that $\varphi:M_{k}\to l^{\infty}(A)/I_{S,\varpi}$ is a
homomorphism. Then, there exists an order zero c.p.c. map
$\Psi=\\{\Psi_{n}\\}:M_{k}\to l^{\infty}(A)$ and a homomorphism $\psi:M_{k}\to
l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}$ such that
$\\{\Psi_{n}(1_{k})\\}$ is a permanent projection lifting of the projection
$\psi(1_{k}),$ $\Phi_{S}\circ\psi=\varphi$ (recall that
$\Phi_{S}:l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}\to
l^{\infty}(A)/I_{{}_{\tau,\varpi}}$ is the quotient map) and
$\Pi_{\varpi}\circ\Psi=\psi.$

###### Proof.

Define a homomorphism $\varphi_{c}:C_{0}((0,1])\otimes M_{k}\to
l^{\infty}(A)/I_{{}_{S,\omega}}$ by $\varphi_{c}(f\otimes
e_{i,j})=f(1)\varphi(e_{i,j})$ for all $f\in C_{0}((0,1])$ and $1\leq i,j\leq
1.$ Since $C_{0}((0,1])\otimes M_{k}$ is semi-projective, there exists a
homomorphism $\Psi_{c}:C_{0}((0,1])\otimes M_{k}\to l^{\infty}(A)$ such that
$\pi_{S}\circ\Psi_{c}=\varphi_{c}.$ Define
$\psi_{c}=\Pi_{\varpi}\circ\Psi_{c}:C_{0}((0,1])\otimes M_{k}\to
l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}.$ Then
$\Phi_{S}\circ\psi_{c}=\varphi_{c}.$ We may write
$\Psi_{c}=\\{\widetilde{\Psi}_{c}^{(m)}\\}_{m\in\mathbb{N}},$ where
$\widetilde{\Psi}_{c}^{(m)}:C_{0}((0,1])\otimes M_{k}\to A$ is a homomorphism.

In what follows we will denote by $\imath$ the identity function on $[0,1].$
Put $\\{a_{n}\\}=\Psi_{c}(\imath\otimes 1_{k})\in l^{\infty}(A)_{+}^{\bf 1}$
and $\\{a_{n}^{(1)}\\}=\Psi_{c}(\imath\otimes e_{1,1})\in
l^{\infty}(A)_{+}^{\bf 1}.$ Thus $\pi_{S}(\\{a_{n}\\})=\psi_{c}(\imath\otimes
1_{k})$ and $\pi_{S}(\\{a_{n}^{(1)}\\})=\psi_{c}(\imath\otimes e_{1,1})$
(recall that $\pi_{S}:l^{\infty}(A)\to l^{\infty}(A)/I_{{}_{S,\varpi}}$ is the
quotient map). By Lemma 3.3, there is
$e_{n}^{(1)}\in\mathrm{Her}(f_{1/4}(a_{n}^{(1)}))_{+}^{\bf 1}$ for each $n$
such that $\\{e_{n}^{(1)}\\}$ is a permanent projection lifting of a
projection $q_{1}=\Pi_{\varpi}(\\{e_{n}^{(1)}\\})$ and
$\pi_{S}(\\{e_{n}^{(1)}\\})=\varphi(e_{1,1})=\varphi_{c}(\imath\otimes
e_{1,1}).$ Put $w_{j,1}=\Psi_{c}(f_{1/8}(\imath)\otimes e_{j,1})$ and
$w_{j,1,n}=\widetilde{\Psi}_{c}^{(n)}(f_{1/8}(\imath)\otimes e_{j,1}),$
$n\in\mathbb{N},$ ($2\leq j\leq k$). Let $w_{j,1,n}=v_{j,1,n}|w_{j,1,n}|$ be
the polar decomposition of $w_{j,1,n}$ in $A^{**}.$ Then
$\Pi_{\varpi}(\\{w_{j,1,n}\\}\\{e_{n}^{(1)}\\}\\{w_{j,1,n}^{*}\\})=\Pi_{\varpi}(\\{v_{j,1,n}e_{n}^{(1)}v_{j,1,n}\\})$
is a projection. Then
$\pi_{S}(w_{1,j}\\{e_{n}^{(1)}\\}w_{1,j}^{*})=\varphi(e_{j,j})=\varphi_{c}(\imath\otimes
e_{j,j})$ ($2\leq j\leq k$). Define, for each $n\in\mathbb{N},$

$\displaystyle
e_{n}=e_{n}^{(1)}+\sum_{j=2}^{k}w_{j,1,n}e_{n}^{(1)}w_{j,1,n}^{*}\in\mathrm{Her}(f_{1/8}(a_{n})).$
(e 3.12)

One checks that $q=\Pi_{\varpi}(\\{e_{n}\\})$ is a projection (recall that
$q_{1}$ is a projection). Then one further checks that $\\{e_{n}\\}$ is a
permanent projection lifting of $q$ and
$\pi_{S}(\\{e_{n}\\})=\varphi(1_{k})=\varphi_{c}(\imath\otimes 1_{k}).$ Note
that

$\displaystyle\\{e_{n}\\}\Psi_{c}(f_{1/8}(\imath)\otimes
1_{k})=\\{e_{n}\\}\,\,\,{\rm and}\,\,\,$ (e 3.13)
$\displaystyle\\{e_{n}\\}\Psi_{c}(f_{1/8}(\imath)\otimes
e_{i,j})=\Psi_{c}(f_{1/8}(\imath)\otimes e_{i,j})\\{e_{n}\\},\,\,1\leq i,j\leq
k.$ (e 3.14)

Define an order zero c.p.c. map $\Psi:M_{k}\to l^{\infty}(A)$ by
$\Psi(e_{i,j})=\\{e_{n}^{1/2}\\}\Psi_{c}(f_{1/8}(\imath)\otimes
e_{i,j})\\{e_{n}^{1/2}\\}$ ($1\leq i,j\leq 1$). Define $\psi:M_{k}\to
l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}$ by
$\psi=\Pi_{\varpi}\circ\Psi.$ It is an order zero c.p.c. map. By (e 3.13),
$\psi(1_{k})=q$ which is a projection. Hence $\psi$ is a homomorphism.
Moreover, since $\pi_{S}(\\{e_{n}\\})=\varphi(1_{k}),$ we also have
$\Phi_{S}\circ\psi=\varphi.$

∎

######  Lemma 3.5.

Let $A$ be a non-elementary separable simple $C^{*}$-algebra with ${\rm
Ped}(A)=A$ which has $T$-tracial approximate oscillation zero and let
$S\subset T(A)$ be a compact subset. Suppose that
$\varphi:M_{k}\to(l^{\infty}(A)\cap A^{\prime})/I_{{}_{S,\varpi}}$ is a
homomorphism.

(1) There are order zero c.p.c. maps
$\Psi^{\prime}=\\{\Psi_{n}^{\prime}\\}:M_{k}\to l^{\infty}(A)\cap A^{\prime}$
and $\psi^{\prime}:M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{\overline{T(A)}^{w},\varpi}}$ such that
$\Pi_{\varpi}\circ\Psi^{\prime}=\psi^{\prime}$ and
$\Phi_{S}\circ\psi^{\prime}=\varphi.$

(2) Moreover, there is also an order zero c.p.c. map $\Psi:M_{k}\to
l^{\infty}(A)$ such that $\\{\Psi_{n}(1_{k})\\}$ is a permanent projection
lifting of the projection $q=\Pi_{\varpi}(\\{\Psi_{n}(1_{k})\\}),$
$q\psi^{\prime}(b)=\psi^{\prime}(b)q$ for all $b\in M_{k}$ and $\psi:M_{k}\to
l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}$ defined by
$\psi(b)=q\psi^{\prime}(b)$ for all $b\in M_{k}$ is a homomorphism.

(3) Furthermore, $\Phi_{S}\circ\psi=\varphi$ and $\Pi_{\varpi}\circ\Psi=\psi.$

###### Proof.

The proof is contained in that of Lemma 3.4. Let us repeat some of the same
argument.

Using the fact that $C^{*}$-algebra $C_{0}((0,1])\otimes M_{k}$ is
semiprojective, one obtains an order zero c.p.c. map
$\psi^{\prime}:M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{\overline{T(A)}^{w},\varpi}}$ such that
$\Phi_{S}\circ\psi^{\prime}=\varphi.$ Let
$\Psi^{\prime}=\\{\Psi_{n}^{\prime}\\}:M_{k}\to l^{\infty}(A)\cap A^{\prime}$
be an order zero c.p.c. map lifting of $\psi^{\prime}.$ So (1) follows.

As in the proof of Lemma 3.4, put
$\pi_{\tau}(\\{a_{n}\\})=\psi_{c}(\imath\otimes 1_{k})\in
l^{\infty}(A)_{+}^{\bf 1}.$ There is
$e_{n}\in\mathrm{Her}(f_{1/4}(a_{n}))_{+}^{\bf 1}$ for each $n$ such that
$\\{e_{n}\\}$ is a permanent projection lifting of a projection
$q:=\Pi_{\varpi}(\\{e_{n}\\})$ and $\pi_{S}(\\{e_{n}\\})=\varphi(1_{k}).$ Note
that, as in the proof of Lemma 3.4, we may also require that
$\\{e_{n}\\}\\{f_{1/8}(\Psi_{n}^{\prime}(1_{k}))\\}=\\{e_{n}\\}$ and
$\\{e_{n}\\}\\{(f_{1/8}(\Psi_{n}^{\prime}))(e_{i,j}))=\\{(f_{1/8}(\Psi_{n}^{\prime}))(e_{i,j})\\}\\{e_{n}\\},$
$1\leq i,j\leq k$ (using functional calculus for order zero maps as in the
proof of 3.4). In particular, $q\psi^{\prime}(b)=\psi^{\prime}(b)q$ for all
$b\in M_{k}.$

Define an order zero c.p.c. map $\Psi:M_{k}\to l^{\infty}(A)$ by
$\Psi(e_{i,j})=\\{e_{n}^{1/2}\\}(f_{1/8}(\Psi))(e_{i,j})\\{e_{n}^{1/2}\\}$
($1\leq i,j\leq 1$). Then $\\{\Psi_{n}(1_{k})\\}=\\{e_{n}\\}$ is a permanent
projection lifting of $q.$ Define $\psi:M_{k}\to
l^{\infty}(A)/I_{{}_{\overline{T(A)}^{w},\varpi}}$ by
$\psi=\Pi_{\varpi}\circ\Psi.$ It is an order zero c.p.c. map. Moreover,
$\psi(b)=\psi^{\prime}(b)q$ for all $b\in M_{k}.$ Since $\psi(1_{k})=q,$
$\psi$ is a homomorphism. Moreover, since
$\pi_{S}(\\{e_{n}\\})=\varphi(1_{k}),$ we have $\Phi_{S}\circ\psi=\varphi.$ So
(2) and (3) hold.

∎

######  Lemma 3.6.

Let $A$ be a separable simple $C^{*}$-algebra with nonempty compact $T(A).$
Suppose that (1) $\partial_{e}(T)=Y\cup Z$ and $Y\cap Z=\emptyset,$ (2) $Y$ is
compact, $Z$ is Borel and, (3) $M_{Y}=\overline{{\rm conv}(Y)}$ and
$M_{Z}=\overline{{\rm conv}(Z)}.$

Let $I_{{}_{M_{Z},\varpi}}=\\{\\{x_{n}\\}\in
l^{\infty}(A):\lim_{n\to\varpi}\sup\\{\|x_{n}\|_{{}_{2,\tau}}=0:\tau\in
M_{Z}\\}.$ Suppose that $\\{e^{\lambda}\\}$ is an approximate identity for
$I_{{}_{M_{Z},\varpi}}.$ Then, for any $\varepsilon>0,$ there exists
$\lambda_{0}$ such that

$\displaystyle\|\Pi_{\varpi}(e^{\lambda})-\Pi_{\varpi}((e^{\lambda})^{2})\|<\varepsilon\,\,\,{\rm
and}\,\,\,$ (e 3.15)
$\displaystyle\lim_{n\to\varpi}\sup\\{\tau(1-(e_{n}^{\lambda})^{2}):\tau\in
M_{Y}\\}<\varepsilon$ (e 3.16)

for all $\lambda\geq\lambda_{0},$ where $\\{e_{n}^{\lambda}\\}\in
l^{\infty}(A)_{+}^{\bf 1}$ is a lifting of $e^{\lambda}.$

###### Proof.

For any $\sigma\in(0,1/4),$ by Corollary 2.15, there is
$f_{0}\in\operatorname{Aff}(T(A))$ such that

$\displaystyle 0\leq f_{0}\leq 1,\,\,(f_{0})|_{M_{Z}}=0\,\,\,{\rm
and}\,\,\,(f_{0})|_{M_{Y}}=1-\sigma.$ (e 3.17)

For any $\eta\in(0,\sigma/4),$ then $f_{0}+\eta\cdot 1_{T(A)}$ is a strictly
positive affine continuous function. By Theorem 9.3 of [41]222The unital
condition of Theorem 9.3 of [41] is not required. The proof is based on
Proposition 2.7, Theorem 2.9 and Corollary 6.4 of [14] which do not require
that $A$ is unital. , we obtain a sequence of $\\{a_{n}\\}\in A_{+}^{\bf 1}$
such that

$\displaystyle\tau(a_{n})>1-1/n\,\,\,{\rm for\,\,\,all}\,\,\,\tau\in
M_{Y}\,\,\,{\rm and}\,\,\,\tau(a_{n})<1/n\,\,\,{\rm for\,\,\,all}\,\,\,\tau\in
M_{Z}.$ (e 3.18)

It follows that $a:=\\{a_{n}\\}\in I_{{}_{M_{Z},\mathbb{N}}}.$

Note that
$\lim_{\lambda}\|\Pi_{\varpi}((a-a^{1/2}(e^{\lambda})^{2}a^{1/2})^{1/2})\|=0.$
Write $e^{\lambda}=\\{e_{n}^{\lambda}\\}\in l^{\infty}(A)_{+}^{\bf 1}.$ It
follows that, for any $\varepsilon\in(0,1/4),$ there is $\lambda_{0}$
satisfying the following: for any $\lambda\geq\lambda_{0},$ there exists
${\cal P}\in\varpi$ such that, if $n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{\tau(a_{n}-a_{n}^{1/2}(e_{n}^{\lambda})^{2}a_{n}^{1/2}):\tau\in
T(A)\\}<(\varepsilon/8)^{2}.$ (e 3.19)

Since $(e^{\lambda})^{1/2}\in I_{{}_{M_{Z},\varpi}},$ without loss of
generality, we may assume that, for all $n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{\tau(e_{n}^{\lambda}):\tau\in
M_{Z}\\}<(\varepsilon/4)^{2}.$ (e 3.20)

We may assume that $1/n<(\varepsilon/8)^{2}$ for all $n\in{\cal P}_{\lambda}.$
We may also assume that, for all $n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{\tau(a_{n}):\tau\in
M_{Y}\\}>1-(\varepsilon/8)^{2}\,\,\,{\rm and}\,\,\,\sup\\{\tau(a_{n}):\tau\in
M_{Z}\\}<(\varepsilon/8)^{2}.$ (e 3.21)

By (e 3.19) and (e 3.21), we have, for $n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{1-\tau(a_{n}^{1/2}(e_{n}^{\lambda})^{2}a_{n}^{1/2}):\tau\in
M_{Y}\\}<(\varepsilon/4)^{2}.$ (e 3.22)

Hence, for $n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{1-\tau((e_{n}^{\lambda})^{2}):\tau\in
M_{Y}\\}\leq\sup\\{1-\tau(a_{n}^{1/2}(e_{n}^{\lambda})^{2}a_{n}^{1/2}):\tau\in
M_{Y}\\}<(\varepsilon/4)^{2}.$ (e 3.23)

It follows that, for all $n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{\tau(e_{n}^{\lambda}-(e_{n}^{\lambda})^{2}):\tau\in
M_{Y}\\}<(\varepsilon/4)^{2}.$ (e 3.24)

We also have, by (e 3.20), for all $n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{\tau(e_{n}^{\lambda}-(e_{n}^{\lambda})^{2}):\tau\in
M_{Z}\\}<(\varepsilon/2)^{2}.$ (e 3.25)

Since $T$ is the convex hull of $M_{Y}$ and $M_{Z},$ we have, for all
$n\in{\cal P}_{\lambda},$

$\displaystyle\sup\\{\tau(e_{n}^{\lambda}-(e_{n}^{\lambda})^{2}):\tau\in
T(A)\\}<(\varepsilon/2)^{2}.$ (e 3.26)

Hence (for each $\lambda\geq\lambda_{0}$)

$\displaystyle\|\Pi_{\varpi}(e^{\lambda}-(e^{\lambda})^{2})\|<\varepsilon.$ (e
3.27)

This and (e 3.23) hold for any $\lambda\geq\lambda_{0}.$ The lemma follows. ∎

######  Lemma 3.7.

Let $A$ be a separable simple $C^{*}$-algebra with nonempty compact $T(A)$
which satisfies condition (C). Let
$\partial_{e}(T(A))=\cup_{n=1}^{\infty}X_{n},$ where $\\{X_{n}\\}$ satisfies
(1) and (2) in 2.12. Suppose that
$K=M_{\partial_{e}(T)\setminus\cup_{i=1}^{J}X_{i+l}}$ for some integer
$J,l\in\mathbb{N}.$ Then, for any separable $C^{*}$-subalgebra of $B\subset
l^{\infty}(A),$ there exists a sequence
$\\{e_{n}\\}\in(I_{{}_{K,\varpi}})_{+}^{\bf 1}$ satisfying the following:

(1)
$\lim_{n\to\infty}\|{[b_{n},\,e_{n}]}\|=0\,\,\,\text{for\,\,\,all}\,\,\,\\{b_{n}\\}\in
B,$

(2) $\Pi_{\varpi}(\\{e_{n}\\})$ is a projection,

(3) $\lim_{n\to\varpi}\|c_{n}e_{n}-c_{n}\|_{{}_{2,QT(A)}}=0,$ if
$c=\\{c_{n}\\}\in B\cap I_{{}_{K,\varpi}}.$

(4) $\lim_{n\to\varpi}\sup\\{1-\tau(e_{n}^{2}):\tau\in
M_{\sum_{i=1}^{J}X_{i+l}}\\}=0.$

###### Proof.

Note that, by 2.12, $K$ is compact (as a closed subset of $T(A)$). Using a
quasi-central approximate identity of $I_{{}_{K,\varpi}},$ we obtain a
sequence
$\\{\\{e_{n}^{(m)}\\}_{n\in\mathbb{N}}\\}\subset(I_{{}_{K,\varpi}})_{+}^{\bf
1}$ such that (in $l^{\infty}(A)$)

$\displaystyle\lim_{m\to\infty}\|e^{(m)}b-be^{(m)}\|=0\,\,\,{\rm
for\,\,\,all}\,\,\,b\in B\,\,\,{\rm and}\,\,\,$ (e 3.29)
$\displaystyle\lim_{m\to\infty}\|e^{(m)}c-c\|=0\,\,\,{\rm
for\,\,\,all}\,\,\,c\in B\cap I_{{}_{K,\varpi}},$

where $e^{(m)}=\\{e_{n}^{(m)}\\}_{n\in\mathbb{N}}.$ Moreover, for any
$\varepsilon>0,$ by Lemma 3.6, there is $m_{0}\in\mathbb{N}$ such that, for
$m\geq m_{0},$

$\displaystyle\lim_{n\to\varpi}\|e_{n}^{(m)}-(e_{n}^{(m)})^{2}\|_{2,T(A)}<\varepsilon/2\,\,\,{\rm
and}\,\,\,$ (e 3.30)
$\displaystyle\lim_{n\to\varpi}\sup\\{1-\tau((e_{n}^{(m)})^{2}):\tau\in
M_{\sum_{i=1}^{J}X_{i+l}}\\}<\varepsilon/2.$ (e 3.31)

We will use Kirchberg’s $\varepsilon$-test (we will use the version of Lemma
3.1 of [35]). Let $\\{\\{b_{n}^{(k)}\\}_{n\in\mathbb{N}}\\}$ be a dense
sequence of $B$ and $\\{b_{n}^{(2k)}\\}$ be a dense sequence of $B\cap
I_{{}_{K,\varpi}}.$ Let $Y_{n}=A_{+}^{\bf 1},$ $n\in\mathbb{N}$ (as $X_{n}$ in
Lemma 3.1 [35]). Define, for each $x\in Y_{n},$

$\displaystyle f_{n}^{(1)}(x)$ $\displaystyle=$
$\displaystyle\|x\|_{2,K}+\sup\\{1-\tau(x^{2}):\tau\in
M_{\sum_{i=1}^{J}X_{i+l}}\\},\,\,\,{\rm and}\,\,\,$ (e 3.32) $\displaystyle
f_{n}^{(k+1)}(x)$ $\displaystyle=$
$\displaystyle\|xb_{n}^{(k)}-b_{n}^{(k)}x\|+\|xb_{n}^{(2k)}-b_{n}^{(2k)}\|+\|x-x^{2}\|_{2,T(A)},\,\,k\in\mathbb{N}.$
(e 3.33)

We have that, since $\\{e_{n}^{(m)}\\}_{n\in\mathbb{N}}\in I_{{}_{K,\varpi}},$
for each $m\in\mathbb{N},$

$\displaystyle\lim_{n\to\varpi}\|e_{n}^{(m)}\|_{2,K}=0\,\,\,{\rm
for\,\,\,all}\,\,\,m.$ (e 3.34)

By (e 3.31), (e 3.29), (e 3.29) and (e 3.30), if $\varepsilon>0$ is given, by
choosing $x_{n}=e_{n}^{(m)}$ for some large $m\geq m_{0}$), we obtain

$\displaystyle\lim_{n\to\varpi}f_{n}^{(k)}(x_{n})<\varepsilon,\,\,1\leq k\leq
K.$ (e 3.35)

Thus, by Kirchberg’s $\varepsilon$-test (see Lemma 3.1 of [35]), we obtain
$e=\\{e_{n}\\}\in l^{\infty}(A)_{+}^{\bf 1}$ such that

$\displaystyle\lim_{n\to\varpi}f_{n}^{(k)}(e_{n})=0\,\,\,{\rm
for\,\,\,all}\,\,\,k\in\mathbb{N}.$ (e 3.36)

By $\lim_{n\to\varpi}f_{n}^{(1)}(e_{n})=0,$ we have
$e=\\{e_{n}\\}\in(I_{{}_{K,\varpi}})_{+}^{\bf 1}.$ Moreover

$\displaystyle\lim_{n\to\varpi}\sup\\{1-\tau(e_{n}^{2}):\tau\in
M_{\sum_{i=1}^{J}X_{i+l}}\\}=0\,\,\,{\rm
and}\,\,\,\lim_{n\to\varpi}\|e_{n}-(e_{n})^{2}\|_{2,T(A)}=0.$ (e 3.37)

Hence $\Pi_{\varpi}(\\{e_{n}\\})$ is a projection. Furthermore, since
$\\{b^{(k)}\\}$ is dense in $B$ and $\\{b^{(2k)}\\}$ is dense in $B\cap
I_{{}_{K,\varpi}},$ respectively, we conclude that

$\displaystyle\lim_{n\to\varpi}\|b_{n}e_{n}-e_{n}b_{n}\|=0\,\,\,{\rm
for\,\,\,all}\,\,\,b=\\{b_{n}\\}\in B\,\,\,{\rm and}\,\,\,$ (e 3.38)
$\displaystyle\lim_{n\to\varpi}\|c_{n}e_{n}-c_{n}\|=0\,\,\,{\rm
for\,\,\,all}\,\,\,c=\\{c_{n}\\}\in B\cap I_{{}_{K,\varpi}}.$ (e 3.39)

The lemma the follows. ∎

######  Lemma 3.8.

Let $A$ be a separable simple $C^{*}$-algebra with nonempty compact $T(A)$
which satisfies condition (C). Let
$\partial_{e}(T(A))=\cup_{n=1}^{\infty}X_{n},$ where $\\{X_{n}\\}$ satisfies
(1) and (2) in 2.12. Let $K_{m}=\cup_{i=1}^{m}X_{i},$
$Y_{k}=\cup_{j=1}^{k}X_{j+m}$ and $Z_{k}=\partial_{e}(T)\setminus Y_{k},$
$k\in\mathbb{N}.$ Suppose that $\\{b_{n}\\}\in(I_{M_{K_{m}}})_{+}^{\bf 1}.$
Then, for any $\varepsilon>0,$ there is ${\cal P}\in\varpi$ satisfying the
following: for any $n\in{\cal P},$ there exists $m(n)\in\mathbb{N}$ such that,
for all $k\geq m(n),$

$\displaystyle\|b_{n}\|_{{}_{2,M_{Z_{k}}}}<\varepsilon$ (e 3.40)

###### Proof.

Fix $\varepsilon\in(0,1/2).$ If the lemma is false, then no ${\cal
P}\in\varpi$ satisfies the said property. However, since
$\\{b_{n}\\}\in(I_{M_{K_{m}}})_{+}^{\bf 1},$ there is ${\cal P}\in\varpi$ such
that, for any $n\in{\cal P},$

$\displaystyle\|b_{n}\|_{{}_{2,K_{m}}}<\varepsilon/4.$ (e 3.41)

Note that ${\cal P}\cap\\{m>n:m\in\mathbb{N}\\}\in\varpi.$ Since the lemma is
assumed to be false, for some $n_{0}\in{\cal P},$ there must be a sequence
$\\{I(n_{0},j)\\}_{j\in\mathbb{N}}$ of integers ($I(n_{0},j)\to\infty$ as
$j\to\infty$) such that

$\displaystyle\|b_{n_{0}}\|_{{}_{2,M_{Z_{l(n_{0},j)}}}}\geq\varepsilon.$ (e
3.42)

We obtain $\tau_{I(n_{0},j)}\in M_{Z_{I(n_{0},j)}}$ such that

$\displaystyle\|b_{n_{0}}\|_{{}_{2,\tau_{I(n_{0},j)}}}\geq\varepsilon.$ (e
3.43)

Since $T(A)$ is compact, we may assume that $\tau_{I(n_{0},j)}\to\tau\in T(A)$
(as $j\to\infty$). Then, by Proposition 2.16, $\tau\in M_{X_{1}}\subset
M_{K_{m}}.$ It follows that

$\displaystyle\|b_{n_{0}}\|_{{}_{2,\tau}}\geq\varepsilon.$ (e 3.44)

A contradiction (to (e 3.41)). ∎

We now combine an idea of Sato and the proof of Theorem 3.3 of [35] (see also
Proposition 3.8 of [25]) with the notion of tracial approximate oscillation
zero.

######  Theorem 3.9.

Let $A$ be a separable non-elementary simple $C^{*}$-algebra with nonempty
compact $T(A)$ satisfying condition (C) such that
$\partial_{e}(T(A))=\cup_{n=1}^{\infty}X_{n}$ which satisfies (1) and (2) in
2.12. Suppose that $A$ has $T$-tracial approximate oscillation zero. Let
$S:=\cup_{j=1}^{m}X_{n}.$ Suppose also that
$\varphi:M_{N}\to(l^{\infty}(A)\cap A^{\prime})/I_{{}_{S,\varpi}}$ is a
homomorphism (for some $N\in\mathbb{N}$).

Then there exists a homomorphism $\psi:M_{N}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{T(A),\varpi}}$ such that $\Phi_{S}\circ\psi=\varphi.$

###### Proof.

Using the fact that $C^{*}$-algebra $C_{0}((0,1])\otimes M_{N}$ is
semiprojective, one obtains an order zero c.p.c. map
$\psi^{\prime}:M_{N}\to(l^{\infty}(A)\cap A^{\prime})/I_{{}_{T(A),\varpi}}$
such that $\Phi_{S}\circ\psi^{\prime}=\varphi.$ Let $\\{a_{n}\\}\in
l^{\infty}(A)\cap A^{\prime}$ be such that
$\Pi_{\varpi}(\\{a_{n}\\})=\psi^{\prime}(1_{N}).$ It follows from Lemma 3.5
that there is a permanent projection lifting $\\{e_{n}\\}\in
l^{\infty}(A)_{+}^{\bf 1}$ of a projection $q\in
l^{\infty}(A)/I_{{}_{T(A),\varpi}}$ such that $\Phi_{S}(q)=\varphi(1_{N}).$
Moreover it follows that $e_{n}\in\mathrm{Her}(f_{1/8}(a_{n}))_{+}^{\bf 1}.$
Since $\\{e_{n}\\}$ is a permanent projection lifting, by replacing $e_{n}$ by
$e_{n}^{1/m(n)}$ for some $m(n)\in\mathbb{N},$ we may assume that, for some
$1/2>\delta_{1}>0,$

$\displaystyle d_{\tau}(e_{n})-f_{\delta_{1}}(e_{n})<1/2^{n}\,\,\,{\rm
for\,\,\,all}\,\,\,\tau\in\overline{T(A)}^{w}\,\,\,{\rm
and}\,\,\,n\in\mathbb{N}.$ (e 3.45)

Define $\psi_{c}^{\prime}:C_{0}((0,1])\otimes M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{T(A),\varpi}}$ by $\psi^{\prime}_{c}(\imath\otimes
e_{i,j})=\psi^{\prime}(e_{i,j})$ ($1\leq i,j\leq k$). Define $\psi:M_{k}\to
l^{\infty}(A)/I_{{}_{T(A),\varpi}}$ by
$\psi(e_{i,j})=q\psi_{c}^{\prime}(f_{1/8}(\imath)\otimes e_{i,j})q.$ The proof
of Lemma 3.5 shows that $\psi^{\prime\prime}$ is a homomorphism and
$\psi^{\prime\prime}(1_{k})=\Pi_{\varpi}(\\{e_{n}\\}).$

We will modify $\\{e_{n}\\}$ to obtain a central sequence.

Let $B$ be the separable $C^{*}$-subalgebra of $l^{\infty}(A)$ generated by
$\iota(A)$ and $\\{e_{n}\\}.$ Let $\\{a_{k}\\}$ be a dense sequence of $A^{\bf
1}.$ Put $b_{n}^{(k)}=e_{n}a_{k}-a_{k}e_{n}.$ Since
$q=\Pi_{{}_{S,\varpi}}(\\{e_{n}\\})$ and $\Phi_{S}(q)=\pi(1_{N}),$ for each
$k\in\mathbb{N},$ $\\{b_{n}^{(k)}\\}_{n\in\mathbb{N}}\in I_{{}_{S,\varpi}}.$
Fix $\varepsilon\in(0,1/2)$ and $K\in\mathbb{N}.$ By Lemma 3.8, there is
${\cal P}\in\varpi,$ satisfying the following: for any $n\in{\cal P},$ there
is $m(n)>m+1$ such that

$\displaystyle\|b_{n}^{(k)}\|_{{}_{2,M_{Z_{n}}}}<\varepsilon/4,\,\,1\leq k\leq
K,$ (e 3.46)

where $Z_{n}=\cup_{i=m(n)}^{\infty}X_{i},$ $n\in\mathbb{N}.$ Put
$F_{n}=\cup_{i=m+1}^{m(n)-1}X_{i}$ and $Y_{n}=\partial_{e}(T(A))\setminus
F_{n},$ ($n\in\mathbb{N}$). We may also write $Y_{n}=S\sqcup Z_{n},$
$n\in\mathbb{N}.$ Hence $I_{{}_{M_{Y_{n}},\varpi}}\subset I_{{}_{S,\varpi}}.$
Since $\overline{{\rm conv}(S)}=M_{S},$ by condition (C), we have
$I_{{}_{S,\varpi}}=I_{{}_{M_{S},\varpi}}.$ For $n\not\in{\cal P}$ and
$n<\min\\{l:l\in{\cal P}\\},$ define $Z_{n}=X_{m+1}.$ If
$n_{p}=\max\\{l<n:l\in{\cal P}\\},$ define $Z_{n}=Z_{n_{p}}.$ Then, for
$n\not\in{\cal P},$ define $F_{n}$ and $Y_{n}$ accordingly.

By Lemma 3.7, there is $\\{p_{l}^{(n)}\\}_{l\in\mathbb{N}}\in
I_{{}_{M_{Y_{n}},\varpi}}$ such that $\Pi_{\varpi}(\\{p_{l}^{(n)}\\})$ is a
projection and ($1\leq k\leq K$)

$\displaystyle\lim_{l\to\varpi}\|a_{k}p_{l}^{(n)}-p_{l}^{(n)}a_{k}\|=0,$ (e
3.47) $\displaystyle\lim_{l\to\varpi}\sup\\{1-\tau((p_{l}^{(n)})^{2}):\tau\in
M_{F_{n}}\\}=0\,\,\,{\rm and}\,\,\,$ (e 3.48)
$\displaystyle\lim_{l\to\varpi}\|p_{l}^{(n)}-(p_{l}^{(n)})^{2}\|_{{}_{2,T(A)}}=0.$
(e 3.49)

Define $d_{l}^{(1,n,k)}=b_{n}^{(k)}p_{l}^{(n)}$ and
$d_{l}^{(2,n,k)}=b_{n}^{(k)}(1-p_{l}^{(n)}),$ $l\in\mathbb{N}.$ Then
$\\{d_{l}^{(1,n,k)}\\}_{l\in\mathbb{N}}\in I_{{}_{M_{Y_{n}},\varpi}}$ and, by
(e 3.48), for $\leq k\leq K,$

$\displaystyle\lim_{l\to\varpi}\|d_{l}^{(2,n,k)}\|_{{}_{2,M_{F_{n}}}}\leq
2\lim_{l\to\varpi}\sup\\{\tau(1-p_{l}^{(n)}):\tau\in M_{F_{n}}\\}=0.$ (e 3.50)

Note that $\\{d_{l}^{(1,n,k)}\\}_{l\in\mathbb{N}}\in
I_{{}_{M_{Y_{n}},\varpi}}.$

By Lemma 3.7, for each fixed $n,$ there is
$\\{q_{l}^{(n)}\\}_{l\in\mathbb{N}}\in(I_{{}_{M_{Y_{n}},\varpi}})_{+}^{\bf 1}$
such that, for $1\leq k\leq K,$

$\displaystyle\lim_{l\to\infty}\|q_{l}^{(n)}\|_{{}_{M_{Y_{n}},\varpi}}=0,$ (e
3.51)
$\displaystyle\lim_{l\to\infty}\|[q_{l}^{(n)},\,a_{k}]\|=0,\,\,\lim_{l\to\infty}\|[q_{l}^{(n)},\,e_{n}]\|=0,$
(e 3.52)
$\displaystyle\lim_{l\to\varpi}\|q_{l}^{(n)}-(q_{l}^{(n)})^{2}\|_{{}_{2,T(A)}}=0\,\,\,{\rm
and}\,\,\,$ (e 3.53)
$\displaystyle\lim_{l\to\varpi}\|d_{l}^{(1,n,k)}q_{l}^{(n)}-d_{l}^{(1,n,k)}\|_{{}_{2,T(A)}}=0.$
(e 3.54)

We estimate that, if $n\in{\cal P},$ by (e 3.54) and (e 3.50), for $1\leq
k\leq K,$

$\displaystyle\lim_{l\to\varpi}\|b_{n}^{(k)}q_{l}^{(n)}-b_{n}^{(k)}\|_{{}_{2,M_{F_{n}}}}\leq\lim_{l\to\varpi}\|d_{l}^{(1,n,k)}q_{l}^{(n)}-d_{l}^{(1,n,k)}\|_{{}_{2,T(A)}}$
(e 3.55) $\displaystyle\hskip
72.26999pt+\lim_{l\to\varpi}\|d_{l}^{(2,n,k)}q_{l}^{(n)}-d_{2}^{(2,n,k)}\|_{{}_{2,M_{F_{n}}}}=0.$
(e 3.56)

Also, by (e 3.54) and (e 3.46), if $n\in{\cal P},$ we have that, for $1\leq
k\leq K,$

$\displaystyle\lim_{l\to\varpi}\|b_{n}^{(k)}q_{l}^{(n)}-b_{n}^{(k)}\|_{{}_{2,M_{Z_{n}}}}\leq\lim_{l\to\varpi}\|d_{l}^{(1,n,k)}q_{l}^{(n)}-d_{l}^{(1,n,k)}\|_{{}_{2,T(A)}}$
(e 3.57) $\displaystyle\hskip
72.26999pt+\lim_{l\to\varpi}\|d_{l}^{(2,n,k)}q_{l}^{(n)}-d_{2}^{(2,n,k)}\|_{{}_{2,M_{Z_{n}}}}<\varepsilon/4.$
(e 3.58)

Since $b_{n}^{(k)}\in I_{{}_{S,\varpi}},$ we also assume that, for all
$n\in{\cal P},$

$\displaystyle\|b_{n}^{(k)}\|_{{}_{2,S}}<\varepsilon/4,\,\,{\rm
whence}\,\,\|b_{n}^{(k)}q_{l}^{(n)}-b_{n}^{(k)}\|_{{}_{2,,S}}<\varepsilon/2.$
(e 3.59)

Note that $T(A)$ is the convex hull of $S,M_{F_{n}}$ and $M_{Z_{n}},$ by
condition (C). We obtain, combining (e 3.56), (e 3.58) and (e 3.59), that, for
any $n\in{\cal P},$

$\displaystyle\lim_{l\to\varpi}\|b_{n}^{(k)}q_{l}^{(n)}-b_{n}^{(k)}\|_{{}_{2,T(A)}}<\varepsilon/2.$
(e 3.60)

Thus, combining with (e 3.51), (e 3.52), (e 3.53), and (e 3.60), for each
$n\in{\cal P},$ there exists $l(n)\in\mathbb{N}$ such that, for $1\leq k\leq
K,$

$\displaystyle\|q_{l(n)}^{(n)}\|_{{}_{2,S}}<1/n^{2},\,\,\|[q_{l(n)}^{(n)},\,a_{k}]\|<1/n^{2}\,\,\,{\rm
and}\,\,\,\|[q_{l(n)}^{(n)},\,e_{n}]\|<1/n^{2},$ (e 3.61)
$\displaystyle\|q_{l(n)}^{(n)}-(q_{l(n)}^{(n)})^{2}\|_{{}_{2,T(A)}}<1/n^{2}\,\,\,{\rm
and}\,\,\,\|b_{n}^{(k)}q_{l(n)}^{(n)}-b_{n}^{(k)}\|_{{}_{2,T(A)}}<\varepsilon/2.$
(e 3.62)

We will apply Kirchberg’s $\varepsilon$-test. Let $W_{n}=A_{+}^{\bf 1}$ (as
$X_{n}$ in Lemma 3.1 of [35]), $n\in\mathbb{N}.$ Define

$\displaystyle f_{n}^{(1)}(x)$ $\displaystyle=$
$\displaystyle\|x\|_{{}_{2,S}}\,\,\,{\rm{and,\,\,\,for}}\,\,k\in N,$
$\displaystyle f_{n}^{(k+1)}$ $\displaystyle=$
$\displaystyle\|xa_{k}-a_{k}x\|+\|xe_{n}-e_{n}x\|+\|(1-x)b_{n}^{(k)}(1-x)\|_{{}_{2,T(A)}}+\|x-x^{2}\|_{{}_{2,T(A)}}.$

By (e 3.61) and (e 3.62), for any $\varepsilon>0$ and any $1\leq k\leq K,$
choosing $q_{l(n)}^{(n)}\in W_{n},$ $n\in\mathbb{N},$ as above, we have that

$\displaystyle\lim_{n\to\varpi}f_{n}^{(k)}(q_{l(n)}^{(n)})<\varepsilon.$ (e
3.63)

Applying Kirchberg’s $\varepsilon$-test (see Lemma 3.1 of [35]), we obtain
$c=\\{c_{n}\\}\in l^{\infty}(A)_{+}^{\bf 1}$ such that

$\displaystyle\lim_{n\to\varpi}f_{n}^{(k)}(c_{n})=0\,\,\,{\rm
for\,\,\,all}\,\,\,k\in\mathbb{N}.$ (e 3.64)

Put ${\bar{c}}=\Pi_{\varpi}(\\{c_{n}\\}).$ Then (since
$\lim_{n\to\varpi}f_{n}^{(1)}(c_{n})=0$) we have that $\\{c_{n}\\}\in
I_{{}_{S,\varpi}}$ and ${\bar{c}}=\Pi_{\varpi}(\\{c_{n}\\})$ is a projection
(as $\lim_{n\to\varpi}\|c_{n}-c_{n}^{2}\|_{{}_{2,T(A)}}=0$). Since
$\\{a_{k}\\}$ is dense in $A,$ we conclude that $\\{c_{n}\\}\in A^{\prime}$
(since $\lim_{n\to\varpi}\|c_{n}a_{k}-a_{k}c_{n}\|=0$) and

$\displaystyle{\bar{c}}\psi^{\prime\prime}(1_{N})=\psi^{\prime\prime}(1_{N}){\bar{c}}\,\,\,\,\,\,\,\,\,({\rm
since}\,\,\lim_{n\to\varpi}\|[c_{n},e_{n}]\|=0),$ (e 3.66)
$\displaystyle(1-{\bar{c}})\psi^{\prime\prime}(1_{N})(1-{\bar{c}})\Pi_{\varpi}(\iota(a))=\Pi_{\varpi}(\iota(a))(1-{\bar{c}}))\psi^{\prime\prime}(1_{N})(1-{\bar{c}})\,\,\,{\rm
for\,\,\,all}\,\,\,a\in A$

(for the last equality, recall that $b_{n}^{(k)}=e_{n}a_{k}-a_{k}e_{n}$ and
$\lim_{n\to\varpi}\|(1-c_{n})b_{n}^{(k)}(1-c_{n})\|_{{}_{2,T(A)}}=0$).

Let ${\bar{e}}_{n}=(1-c_{n})e_{n}(1-c_{n}),$ $n\in\mathbb{N},$ and
${\bar{q}}=\Pi_{\varpi}(\\{\bar{e}_{n}\\}).$ By (e 3.66),
$q\bar{c}={\bar{c}}q.$ Since both $q$ and ${\bar{c}}$ are projections, so is
${\bar{q}}.$ We also have ${\bar{q}}\leq q.$ Moreover,
$\Pi_{\varpi}(\\{\bar{e}_{n}\\})=(1-{\bar{c}})\psi^{\prime\prime}(1_{N})(1-{\bar{c}})$
and $\\{{\bar{e}_{n}}\\}\in\Pi_{\varpi}(\iota(A))^{\prime}$ (by (e 3.66)).
Define $\psi:M_{N}\to l^{\infty}(A)/I_{{}_{T(A),\varpi}}$ by
$\psi(e_{i,j})={\bar{q}}\psi_{c}^{\prime}(f_{1/8}(\imath)\otimes
e_{i,j}){\bar{q}}$ $(1\leq i,j\leq N$). It is an order zero c.p.c. map. Then
(recall that ${\bar{q}}$ is a projection and ${\bar{q}}\leq q$), as in the
first paragraph of this proof, ${\bar{\psi}}(1_{N})={\bar{q}}.$ Since
$\\{\bar{e}_{n}\\}\in\Pi_{\varpi}(\iota(A))^{\prime}$ and ${\bar{q}}$ is a
projection, $\psi$ is a homomorphism. Recall that $\psi_{c}^{\prime}$ maps
into $(l^{\infty}(A)\cap A^{\prime})/I_{{}_{T(A),\varpi}}$ and
$\\{\bar{e}_{n}\\}\in\Pi_{\varpi}(\iota(A))^{\prime},$ the map $\psi$ is
actually a homomorphism from $M_{N}$ into
$(l^{\infty}(A)/I_{{}_{T(A),\varpi}})\cap\Pi_{\varpi}(\iota(A))^{\prime}.$ By
Central Surjectivity (see Lemma 1.2 of [11], also Proposition 4.5 (iii) and
Proposition 4.6 of [35]),
$(l^{\infty}(A)/I_{{}_{T(A),\varpi}})\cap\Pi_{\varpi}(\iota(A))^{\prime}=(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{T(A),\varpi}}.$ Therefore $\psi$ is a homomorphism from
$M_{N}$ to $(l^{\infty}(A)\cap A^{\prime})/I_{{}_{T(A),\varpi}}.$ Since
$\\{c_{n}\\}\in I_{{}_{S,\varpi}},$ we have $\Phi_{S}\circ\psi=\varphi.$ The
lemma then follows.

∎

######  Definition 3.10.

Let $A$ be a separable simple $C^{*}$-algebra with $A={\rm Ped}(A),$
$S\subset\overline{T(A)}^{w}$ be a compact subset and $\varphi:M_{k}\to
l^{\infty}(A)\cap A^{\prime}/I_{{}_{S,\varpi}}$ be an order zero c.p.c. map.
We say that $\varphi$ has property (Os), if there exists an order zero c.p.c.
map $\Psi=\\{\widetilde{\psi}_{n}\\}:M_{k}\to l^{\infty}(A)\cap A^{\prime}$
such that $\\{\widetilde{\psi}_{n}(1_{k})\\}$ is a permanent projection
lifting of $\psi(1_{k}),$ and, for any $\varepsilon>0$ there exist $\delta>0$
and ${\cal Q}\in\varpi$ such that

$\displaystyle
d_{\tau}(\widetilde{\psi}_{n}(1_{k}))-\tau(f_{\delta}(\widetilde{\psi}_{n}(1_{k})))<\varepsilon\,\,\,\text{for\,\,\,all}\,\,\,\tau\in\overline{T(A)}^{w}\,\,\,\text{and}\,\,\,n\in{\cal
Q}.$ (e 3.67)

######  Proposition 3.11.

Let $A$ be a separable non-elementary simple $C^{*}$-algebra with $A={\rm
Ped}(A)$ and $T(A)\not=\emptyset.$ Suppose that
$\varphi:M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{\overline{T(A)}^{w},\varpi}}$ is a homomorphism. Then
$\varphi$ has property (Os).

###### Proof.

Let $\psi_{c}:C_{0}((0,1])\otimes M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{T(A),\varpi}}$ be defined by $\psi_{c}(g\otimes
e_{i,j})=g(1)\psi(e_{i,j})$ for all $g\in C_{0}((0,1])$ and $1\leq i,j\leq k.$
Since $C_{0}((0,1])\otimes M_{k}$ is semiprojective, there is a homomorphism
$\Psi_{c}=\\{\widetilde{\psi}_{c,n}\\}:C_{0}((0,1])\otimes M_{k}\to
l^{\infty}(A)\cap A^{\prime}$ such that $\Pi_{\varpi}\circ\Psi_{c}=\psi_{c}.$
Note that
$\Pi_{\varpi}(\\{\widetilde{\psi}_{c,n}(\imath\otimes 1_{k})\\})=\psi(1_{k}).$
Define $\Psi=\\{\widetilde{\psi}_{n}\\}:M_{k}\to l^{\infty}(A)\cap A^{\prime}$
by
$\widetilde{\psi}_{n}(e_{i,j})=\widetilde{\psi}_{c,n}(f_{1/4}(\imath)\otimes
e_{i,j})$ for $1\leq i,j\leq k.$ Then $\Psi$ is an order zero c.p.c. map and
$\Pi_{\varpi}\circ\Psi=\varphi.$ Moreover
$\\{\widetilde{\psi}_{n}(1_{k})\\}=\\{f_{1/4}(\widetilde{\psi}_{c,n}(1_{k}))\\}$
is a permanent projection lifting of $\varphi(1_{k})$ (see Proposition 2.22).
The proposition then follows from Proposition 2.22. ∎

######  Corollary 3.12.

Let $A$ be a separable non-elementary simple $C^{*}$-algebra with nonempty
compact $T(A)$ satisfying condition (C) such that
$\partial_{e}(T(A))=\cup_{n=1}^{\infty}X_{n}$ satisfies (1) and (2) in 2.12.
Suppose that $A$ has T-tracial approximate oscillation zero and suppose that
$\varphi:M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{\cup_{i=1}^{n}X_{i},\varpi}}$ (for some $n\in\mathbb{N}$)
is a homomorphism. Then $\varphi$ has property (Os).

###### Proof.

Since $A$ has T-tracial approximate oscillation zero, by Theorem 3.9, there is
a homomorphism $\varphi:M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{T(A),\varpi}}$ such that $\Phi_{S}\circ\psi=\varphi.$ Then
the corollary follows from Proposition 3.11. ∎

##  4 Sums of order zero maps

We fix a free ultrafilter $\varpi\in{\beta}(\mathbb{N})\setminus\mathbb{N}.$

######  Lemma 4.1.

Let $A$ be a simple $C^{*}$-algebra with compact $T(A)\not=\emptyset.$ Suppose
that $a,b\in A_{+}^{\bf 1}\setminus\\{0\\}$ such that

$\displaystyle\tau(a)>1-\varepsilon$ (e 4.1)

for some $\tau\in T(A)$ and $\varepsilon\in(0,1/2).$ Then

$\displaystyle\tau(bab)>\tau(b^{2})-\sqrt{\varepsilon}.$ (e 4.2)

###### Proof.

We have

$\displaystyle\tau(b^{2}-bab)^{2}$ $\displaystyle=$
$\displaystyle\tau(b(1-a)b)^{2}\leq\tau(b^{2})\tau((1-a)b^{2}(1-a))$ (e 4.3)
$\displaystyle<$
$\displaystyle\tau(b^{2})\tau(1-a)=\tau(b^{2})(1-\tau(a))<\varepsilon.$ (e
4.4)

It follows that

$\displaystyle\tau(bab)=\tau(b^{2})-\tau(b^{2}-bab)>\tau(b^{2})-\sqrt{\varepsilon}$
(e 4.5)

as desired. ∎

######  Lemma 4.2.

Let $A$ be a separable amenable simple $C^{*}$-algebra with nonempty compact
$T(A)$ and let $k\in\mathbb{N}.$ Let $\\{S_{n}\\}\subset\partial_{e}(T(A))$ be
an increasing sequence of compact subsets. Suppose that, for each $n,$ there
exists a unital homomorphism $\psi_{n}:M_{k}\to(l^{\infty}(A)\cap
A^{\prime})/I_{{}_{S_{n},\varpi}}$ which has property (Os).

Then, for any $\varepsilon>0$ and any finite subset ${\cal F}\subset A,$ there
exists a sequence of order zero c.p.c. maps $\varphi_{n}:M_{k}\to A$ such that

$\displaystyle\|[\varphi_{n}(b),\,a]\|<\sum_{j=1}^{n}\varepsilon/2^{i+2}\,\,\,\text{for\,\,\,all}\,\,\,a\in{\cal
F}\,\,\,\text{and}\,\,\,b\in M_{k}^{\bf 1},$ (e 4.6)
$\displaystyle\tau(\varphi_{n}(1_{k}))>1-(\varepsilon/2^{n+5})^{2}\,\,\,\text{for\,\,\,all}\,\,\,\tau\in\cup_{j=1}^{n}X_{j},$
(e 4.7)

and, if $\tau(\varphi_{n}(1_{k}))>1-(\sigma/2)^{2}$ for some $\tau\in T(A)$
and $\sigma\in(0,1/2),$ then

$\displaystyle\tau(\varphi_{n+1}(1_{k}))>1-(\varepsilon/2^{n+4})^{2}-\sigma/2\,\,\,\text{for\,\,\,all}\,\,\,n\in\mathbb{N}.$
(e 4.9)

###### Proof.

Since $C_{0}((0,1])\otimes M_{k}$ is semiprojective, for any $\varepsilon>0,$
there is a universal $\eta_{n}>0,$ for each $n\in\mathbb{N}$ satisfying the
following: for any c.p.c. map $L:M_{k}\to B$ (for any $C^{*}$-algebra $B$)
such that

$\displaystyle\|L(a)L(b)\|<\eta_{n}$ (e 4.10)

for any pair of mutually orthogonal elements $a,b\in(M_{k})_{+}^{\bf 1},$
there exists an order zero map $\Phi:M_{k}\to B$ such that

$\displaystyle\|L-\Phi\|<(\varepsilon/2^{2(n+2)})^{2}$ (e 4.11)

(recall the unit ball of $M_{k}$ is compact). Without loss of generality, we
may assume that $\eta_{n}<\varepsilon/2^{3(n+1)}$.

We will construct $\\{\varphi_{n}\\}$ by induction.

For $n=1,$ Let $\Psi_{1}=\\{\psi_{1}^{(m)}\\}:M_{k}\to l^{\infty}(A)\cap
A^{\prime}$ be an order zero c.p.c. map such that
$\Pi_{\varpi}\circ\Psi_{1}=\psi_{1}.$ Since
$\psi_{1}:M_{k}\to(l^{\infty}(A)\cap A^{\prime})/I_{S_{1},\varpi}$ is unital,
there exists ${\cal P}_{1}\in\varpi$ such that, for any $m\in{\cal P}_{1},$

$\displaystyle s(\psi_{1}^{(m)}(1_{k}))>1-(\varepsilon/2^{2+5})^{2}\,\,\,{\rm
for\,\,\,all}\,\,\,s\in S_{1}.$ (e 4.12)

Therefore we obtain $m_{1}\in\mathbb{N}$ such that

$\displaystyle\|[\psi_{1}^{(m_{1})}(b),\,a]\|<\eta_{1}\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}\,\,\,{\rm and}\,\,\,b\in M_{k}^{\bf
1}\,\,\,{\rm and}\,\,\,$ (e 4.13) $\displaystyle
s(\varphi_{1}^{(m_{1})}(1_{k}))>1-(\varepsilon/2^{1+5})^{2}\,\,\,{\rm
for\,\,\,all}\,\,\,s\in S_{1}.$ (e 4.14)

Put $\varphi_{1}=\psi_{1}^{(m_{1})}$ and ${\cal F}_{1}={\cal
F}\cup\varphi_{1}(M_{k}^{\bf 1}).$

Let $\Psi_{2}=\\{\psi_{2}^{(m)}\\}:M_{k}\to l^{\infty}(A)\cap A^{\prime}$ be
an order zero c.p.c. map such that $\Pi_{\varpi}\circ\Psi_{2}=\psi_{2}.$ We
assume, since $\psi_{2}$ has property (Os), there is $\delta_{2}\in(0,1/4)$
and ${\cal P}_{2}\in\varpi$ such that, for all $m\in{\cal P}_{2},$

$\displaystyle\sup\\{d_{\tau}(\psi_{2}^{(m)}(1_{k}))-\tau(f_{\delta_{2}}(\psi_{2}^{(m)}(1_{k}))):\tau\in
T(A)\\}<(\varepsilon/2^{2+8})^{2}.$ (e 4.15)

By (1) of Proposition 2.22, we may assume that, for all $m\in{\cal P}_{2},$

$\displaystyle\sup\\{\tau(\psi_{2}^{(m)}(1_{k}))-\tau(f_{\delta_{2}}(\psi_{2}^{(m)}(1_{k}))\psi_{2}^{(m)}(1_{k})):\tau\in
T(A)\\}<(\varepsilon/2^{2+8})^{2}.$ (e 4.16)

Without loss of generality (recall that $\psi_{2}:M_{k}\to l^{\infty}(A)\cap
A^{\prime}/I_{{}_{S_{2},\varpi}}$ is unital), we may choose $m_{2}\in{\cal
P}_{2}$ such (recall that $\varphi_{1}((M_{k})^{\bf 1})$ is compact). that

$\displaystyle
s(\psi_{2}^{(m_{2})}(1_{k}))>1-(\varepsilon/2^{2+8})^{2}\,\,\,{\rm
for\,\,\,all}\,\,\,s\in S_{2},$ (e 4.17)
$\displaystyle\|[\psi_{2}^{(m_{2})}(b),\,a]\|<\eta_{2}\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}_{1}\,\,\,{\rm and}\,\,\,b\in M_{k}^{\bf 1},$
(e 4.18)
$\displaystyle\|[f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k})),\,a]\|<\eta_{2}\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}_{1}\,\,\,{\rm and}\,\,\,$ (e 4.19)
$\displaystyle\|[(1-f_{\delta_{2}/2}(\psi_{2}^{(m_{2})}(1_{k})))^{1/2},\,a]\|<\eta_{2}\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}_{1}.$ (e 4.20)

Put

$\displaystyle a_{2}=f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))\,\,\,{\rm
and}\,\,\,a_{2}^{\perp}=(1-f_{\delta_{2}/2}(\psi_{2}^{(m_{2})}(1_{k})))^{1/2}.$
(e 4.21)

Let $H_{2}:C_{0}((0,1])\otimes M_{k}\to A$ be a homomorphism such that
$H_{2}(\imath\otimes e_{i,j})=\psi_{2}^{(m_{2})}(e_{i,j})$ ($1\leq i,j\leq
k$). Then, for any $\delta\in(0,1/2),$
$f_{\delta}(\psi_{2}^{(m_{2})}(1_{k}))=H_{2}(f_{\delta}(\imath)\otimes
1_{k}).$ It follows that $a_{2}$ and $a_{2}^{\perp}$ commutes with elements of
the form $\psi_{2}^{(m_{2})}(b)$ for $b\in M_{k}$ (this will be used for (e
4.29)). Define $L_{2}:M_{k}\to A$ by

$\displaystyle L_{2}(b)$ $\displaystyle=$ $\displaystyle
a_{2}^{\perp}\varphi_{1}(b)a_{2}^{\perp}+a_{2}\psi_{2}^{(m_{2})}(b)a_{2}\,\,\,{\rm
for\,\,\,all}\,\,\,b\in M_{k}.$

Then $L_{2}$ is a c.p.c. map.

By (e 4.16), we compute that, for all $\tau\in T(A),$

$\displaystyle\sup\\{\tau(\psi_{2}^{(m_{2})}(1_{k})-f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))\psi_{2}^{(m_{2})}(1_{k})f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))):\tau\in
T(A)\\}$
$\displaystyle\leq\sup\\{\tau(\psi_{2}^{(m_{2})}(1_{k})-f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))\psi_{2}^{(m_{2})}(1_{k})):\tau\in
T(A)\\}+$ $\displaystyle\hskip
14.45377pt\sup\\{\tau(f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))\psi_{2}^{(m_{2})}(1_{k}))-\tau(f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))\psi_{2}^{(m_{2})}(1_{k})f_{\delta}(\psi_{2}^{(m_{2})}(1_{k}))):\tau\in
T(A)\\}$
$\displaystyle<(\varepsilon/2^{2+8})^{2}+\|f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))\|\sup\\{\tau(\psi_{2}^{(m_{2})}(1_{k})-\psi_{2}^{(m_{2})}(1_{k})f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))):\tau\in
T(A)\\}$
$\displaystyle<(\varepsilon/2^{2+8})^{2}+(\varepsilon/2^{2+8})^{2}<(\varepsilon/2^{2+7})^{2}.$
(e 4.22)

Similarly (using (e 4.15)), for all $\tau\in T(A),$

$\displaystyle\sup\\{\tau(f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))-f_{\delta_{2}}(\psi_{2}^{(m_{2})}(1_{k}))\psi_{2}^{(m_{2})}(1_{k})f_{\delta}(\psi_{2}^{(m_{2})}(1_{k})):\tau\in
T(A)\\}<(\varepsilon/2^{2+7})^{2}.$ (e 4.23)

Then, for $\tau\in S_{2},$ by (e 4.22) and (e 4.17),

$\displaystyle\tau(L_{2}(1_{k}))>\tau(a_{2}\psi_{2}^{(m_{2})}(1_{k})a_{2})>\tau(\psi_{2}^{(m_{2})}(1_{k}))-(\varepsilon/2^{2+7})^{2}>1-(\varepsilon/2^{2+6})^{2}.$
(e 4.24)

Moreover, if, for some $\tau\in T(A),$
$\tau(\varphi_{1}(1_{k}))>1-(\sigma/2)^{2},$ then, by Lemma 4.1, (e 4.23) and
(e 4.15),

$\displaystyle\tau(L_{2}(1_{k}))$ $\displaystyle=$
$\displaystyle\tau(a_{2}^{\perp}\varphi_{1}(1_{k})a_{2}^{\perp})+\tau(a_{2}\psi_{2}^{(m_{2})}(1_{k})a_{2})$
(e 4.25) $\displaystyle>$
$\displaystyle\tau((a_{2}^{\perp})^{2})-\sigma/2+\tau(f_{\delta_{2}}(\varphi_{2}^{(m_{2})})\psi_{2}^{(m_{2})}(1_{k})f_{\delta_{2}}(\varphi_{2}^{(m_{2})}(1_{k}))$
$\displaystyle=$
$\displaystyle\tau((1-f_{\delta_{2}/2}(\psi_{2}^{(m_{2})}(1_{k}))+f_{\delta_{2}}(\varphi_{2}^{(m_{2})}(1_{k}))\psi_{2}^{(m_{2})}(1_{k})f_{\delta_{2}}(\varphi_{2}^{(m_{2})}(1_{k}))))-\sigma/2$
$\displaystyle\geq$
$\displaystyle\tau(1-f_{\delta_{2}/2}(\psi_{2}^{(m_{2})}(1_{k}))+f_{\delta_{2}}(\varphi_{2}^{(m_{2})}(1_{k})))-(\varepsilon/2^{2+7})^{2}-\sigma/2$
$\displaystyle>$ $\displaystyle
1-(\varepsilon/2^{2+8})^{2}-(\varepsilon/2^{2+7})^{2}-\sigma/2.$

For any $a\in{\cal F}$ and $b\in M_{k}^{\bf 1},$ by (e 4.20), (e 4.19) and by
(e 4.13),

$\displaystyle
xL_{2}(b)\approx_{2\eta_{2}}a_{2}^{\perp}x\varphi_{1}(b)a_{2}^{\perp}+a_{2}x\psi_{2}^{(m_{2})}(b)a_{2}\approx_{\eta_{1}+3\eta_{2}}L_{2}(b)x.$
(e 4.26)

Suppose that $b,c\in(M_{k})^{\bf 1}_{+}$ such that $bc=cb=0.$ Then, since both
$\varphi_{1}$ and $\psi_{2}^{(m_{2})}$ are order zero maps (see (e 4.19)),

$\displaystyle L_{2}(b)L_{2}(c)$ $\displaystyle=$
$\displaystyle(a_{2}^{\perp}\varphi_{1}(b)a_{2}^{\perp}+a_{2}\psi_{2}^{(m_{2})}(b)a_{2})(a_{2}^{\perp}\varphi_{1}(c)a_{2}^{\perp}+a_{2}\psi_{2}^{(m_{2})}(c)a_{2})$
(e 4.27) $\displaystyle=$ $\displaystyle
a_{2}^{\perp}\varphi_{1}(b)a_{2}^{\perp}a_{2}^{\perp}\varphi_{1}(c)a_{2}^{\perp}+a_{2}\psi_{2}^{(m_{2})}(b)a_{2}^{2}\psi_{2}^{(m_{2})}(c)a_{2}$
(e 4.28) $\displaystyle\approx_{\eta_{2}}$
$\displaystyle(a_{2}^{\perp})^{3}\varphi_{1}(b)\varphi_{1}(c)a_{2}^{\perp}=0.$
(e 4.29)

By the choice of $\eta_{2},$ we obtain an order zero c.p.c. map
$\varphi_{2}:M_{k}\to A$ such that

$\displaystyle\|L_{2}-\varphi_{2}\|<(\varepsilon/2^{2(2+2)})^{2}=(\varepsilon/2^{8})^{2}.$
(e 4.30)

By (e 4.26),

$\displaystyle\|[x,\,\varphi_{2}(b)]\|<\eta_{1}+5\eta_{2}+2(\varepsilon/2^{8})^{2}<\varepsilon/2^{4}\,\,\,{\rm
for\,\,\,all}\,\,\,x\in{\cal F}\,\,\,{\rm and}\,\,\,b\in M_{k}^{\bf 1}.$ (e
4.31)

Therefore, by (e 4.24), for $s\in S_{2},$

$\displaystyle
s(\varphi_{2}(1_{k}))>1-(\varepsilon/2^{6})^{2}-(\varepsilon/2^{8})^{2}>1-(\varepsilon/2^{2+5})^{2}.$
(e 4.32)

If $\tau\in T(A)$ such that $\tau(\varphi_{1}(1_{k}))>1-(\sigma/2)^{2},$ then,
by (e 4.25),

$\displaystyle\tau(\varphi_{2}(1_{k}))>1-(\varepsilon/2^{2+8})^{2}-(\varepsilon/2^{2+7})^{2}-\sigma/2-(\varepsilon/2^{8})^{2}>1-(\varepsilon/2^{2+5})^{2}-\sigma/2.$
(e 4.33)

Suppose that order zero c.p.c. maps
$\varphi_{1},\varphi_{2},...,\varphi_{n}:M_{k}\to A$ have been constructed
which meet the requirements of the lemma.

Let ${\cal F}_{n}={\cal F}\cup\\{\varphi_{n}(b):b\in M_{k}^{\bf 1}\\}.$ Let
$\Psi_{n+1}=\\{\psi_{n+1}^{(m)}\\}:M_{k}\to l^{\infty}(A)\cap A^{\prime}$ be
an order zero c.p.c. map such that $\Pi_{\varpi}\circ\Psi_{n+1}=\psi_{n+1}.$
We assume, since $\psi_{n+1}$ has property (Os), there is
$\delta_{n+1}\in(0,1/4)$ and ${\cal P}_{n+1}\in\varpi$ such that, for all
$m\in{\cal P}_{n+1},$

$\displaystyle\sup\\{d_{\tau}(\psi_{n+1}^{(m)}(1_{k}))-\tau(f_{\delta_{n+1}}(\psi_{n+1}^{(m)}(1_{k}))):\tau\in
T(A)\\}<({\varepsilon\over{2^{n+1+8}}})^{2}.$ (e 4.34)

By (1) of Proposition 2.22, we may assume that

$\displaystyle\sup\\{\tau(\psi_{n+1}^{(m)}(1_{k}))-\tau(f_{\delta_{n+1}}(\psi_{2}^{(m)}(1_{k}))\psi_{2}^{(m)}(1_{k})):\tau\in
T(A)\\}<({\varepsilon\over{2^{n+1+8}}})^{2}.$ (e 4.35)

Without loss of generality (recall that $\psi_{n+1}:M_{k}\to l^{\infty}(A)\cap
A^{\prime}/I_{{}_{S_{n_{1}},\varpi}}$ is unital), we may choose
$m_{n+1}\in{\cal P}_{n+1}$ such that

$\displaystyle
s(\psi_{n+1}^{(m_{n+1})}(1_{k}))>1-(\varepsilon/2^{n+1+8})^{2}\,\,\,{\rm
for\,\,\,all}\,\,\,s\in S_{n+1},$ (e 4.36)
$\displaystyle\|[\psi_{n+1}^{(m_{n+1})}(b),\,a]\|<\eta_{n+1}\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}_{n},$ (e 4.37)
$\displaystyle\|[f_{\delta_{n+1}}(\psi_{n+1}^{(m_{n+1})}(1_{k})),\,a]\|<\eta_{n+1}\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}_{n}\,\,\,{\rm and}\,\,\,$ (e 4.38)
$\displaystyle\|[(1-f_{\delta_{n+1}/2}(\psi_{n+1}^{(m_{n+1})}(1_{k})))^{1/2},\,a]\|<\eta_{n+1}\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}_{n}.$ (e 4.39)

Put

$\displaystyle
a_{n+1}=f_{\delta_{n+1}}(\psi_{n+1}^{(m_{n+1})}(1_{k}))\,\,\,{\rm
and}\,\,\,a_{n+1}^{\perp}=(1-f_{\delta_{n+1}/2}(\psi_{n+1}^{(m_{n+1})}(1_{k})))^{1/2}.$
(e 4.40)

Let $H_{n+1}:C_{0}((0,1])\otimes M_{k}\to A$ be a homomorphism such that
$H_{n+1}(\imath\otimes e_{i,j})=\psi_{2}^{(m_{2})}(e_{i,j})$ ($1\leq i,j\leq
k$). Then, for any $\delta\in(0,1/2),$
$f_{\delta}(\psi_{2}^{(m_{2})}(1_{k}))=H_{n+1}(f_{\delta}(\imath)\otimes
1_{k}).$ It follows that $a_{n+1}$ and $a_{n+1}^{\perp}$ commutes with
elements of the form $\psi_{n+1}^{(m_{n+1})}(b)$ for $b\in M_{k}.$ Define
$L_{n+1}:M_{k}\to A$ by

$\displaystyle L_{n+1}(b)$ $\displaystyle=$ $\displaystyle
a_{n+1}^{\perp}\varphi_{n}(b)a_{n+1}^{\perp}+a_{n+1}\psi_{n+1}^{(m_{n+1})}(b)a_{n+1}\,\,\,{\rm
for\,\,\,all}\,\,\,b\in M_{k}.$

Then $L_{n+1}$ is a c.p.c. map. By (e 4.35), we compute that, for all $\tau\in
T(A),$

$\displaystyle\sup\\{\tau(\psi_{n+1}^{(m_{n+1})}(1_{k}))-a_{n+1}\varphi_{n+1}^{(m_{n+1})}(1_{k})a_{n+1}:\tau\in
T(A)\\}<({\varepsilon\over{2^{n+1+7}}})^{2}\,\,\,{\rm and}\,\,\,$ (e 4.41)
$\displaystyle\sup\\{\tau(f_{\delta_{n+1}}(\psi_{n+1}^{(m_{n+1})}(1_{k})))-a_{n+1}\psi_{n+1}^{(m_{n+1})}(1_{k})a_{n+1}:\tau\in
T(A)\\}<({\varepsilon\over{2^{n+1+7}}})^{2}.$ (e 4.42)

It follows from (e 4.42) that, for all $\tau\in T(A),$

$\displaystyle\tau((a_{n+1}^{\perp})^{2}+a_{n+1}\psi_{n+1}^{(m_{n+1})}(1_{k})a_{n+1})>1-({\varepsilon\over{2^{n+1+7}}})^{2}.$
(e 4.43)

For $\tau\in S_{n+1},$ by (e 4.41) and (e 4.36),

$\displaystyle
s(L_{n+1}(1_{k}))>s(a_{n+1}\psi_{n+1}^{(m_{n+1})}(1_{k})a_{n+1})>s(\psi_{2}^{(m_{2})}(1_{k}))-({\varepsilon\over{2^{n+7}}})^{2}>1-({\varepsilon\over{2^{n+6}}})^{2}.$
(e 4.44)

Moreover, if $\tau(\varphi_{n}(1_{k}))>1-(\sigma/2)^{2}$ for some $\tau\in
T(A),$ then, by Lemma 4.1 and by (e 4.43),

$\displaystyle\tau(L_{n+1}(1_{k}))$ $\displaystyle=$
$\displaystyle\tau(a_{n+1}^{\perp}\varphi_{n}(1_{k})a_{n+1}^{\perp})+\tau(a_{n+1}\psi_{n+1}^{(m_{n+1})}(1_{k})a_{n+1})$
$\displaystyle>$
$\displaystyle\tau((a_{n+1}^{\perp})^{2})-\sigma/2+\tau(a_{n+1}\psi_{2}^{(m_{n+1})}(1_{k})a_{n+1})$
$\displaystyle=$
$\displaystyle\tau((a_{n+1}^{\perp})^{2}+a_{n+1}\psi_{n+1}^{(m_{n+1})}(1_{k})a_{n+1})-\sigma/2$
$\displaystyle>$ $\displaystyle 1-(\varepsilon/2^{n+7})^{2}-\sigma/2.$ (e
4.45)

For any $a\in{\cal F}$ and $b\in M_{k}^{\bf 1},$ by (e 4.39), (e 4.38) and by
inductive assumption,

$\displaystyle xL_{n+1}(b)$ $\displaystyle\approx_{2\eta_{n+1}}$
$\displaystyle
a_{n+1}^{\perp}x\varphi_{n}(b)a_{n+1}^{\perp}+a_{n+1}x\psi_{n+1}^{(m_{n+1})}(b)a_{n+1}$
(e 4.47)
$\displaystyle\approx_{(\sum_{i=1}^{n}\varepsilon/2^{i+2})+3\eta_{n+1}}L_{n+1}(b)x.$

Suppose that $b,c\in(M_{k})^{\bf 1}_{+}$ such that $bc=cb=0.$ Then, since both
$\varphi_{n}$ and $\psi_{n+1}^{(m_{2})}$ are order zero maps (see (e 4.38)),

$\displaystyle L_{n+1}(b)L_{n+1}(c)=$
$\displaystyle(a_{n+1}^{\perp}\varphi_{n}(b)a_{n+1}^{\perp}+a_{n+1}\psi_{n+1}^{(m_{n+1})}(b)a_{n+1})(a_{n+1}^{\perp}\varphi_{n}(c)a_{n+1}^{\perp}+a_{n+1}\psi_{n+1}^{(m_{n+1})}(c)a_{n+1})$
$\displaystyle=a_{n+1}^{\perp}\varphi_{n}(b)a_{n+1}^{\perp}a_{n+1}^{\perp}\varphi_{n}(c)a_{n+1}^{\perp}+a_{n+1}\psi_{n+1}^{(m_{n+1})}(b)a_{n+1}^{2}\psi_{n+1}^{(m_{n+1})}(c)a_{n+1}$
$\displaystyle\approx_{\eta_{n+1}}(a_{n+1}^{\perp})^{3}\varphi_{n}(b)\varphi_{n}(c)a_{n+1}^{\perp}=0.$
(e 4.48)

By the choice of $\eta_{n+1},$ we obtain an order zero c.p.c. map
$\varphi_{n+1}:M_{k}\to A$ such that

$\displaystyle\|L_{n+1}-\varphi_{n+1}\|<(\varepsilon/2^{2(n+1+2)})^{2}.$ (e
4.49)

Then, by (e 4.47), for all $x\in{\cal F}$ and $b\in{\cal F},$

$\displaystyle\|[x,\,\varphi_{n+1}(b)]\|<\sum_{i=1}^{n}\varepsilon/2^{i+2}+5\eta_{n+1}+2(\varepsilon/2^{2(n+1+2)})^{2}<\sum_{i=1}^{n+1}\varepsilon/2^{i+2}.$
(e 4.50)

Also, by (e 4.44), for $s\in S_{n+1},$

$\displaystyle
s(\varphi_{n+1}(1_{k}))>1-({\varepsilon\over{2^{n+6}}})^{2}-(\varepsilon/2^{2(n+1+2)})^{2}>1-(\varepsilon/2^{n+5})^{2}.$
(e 4.51)

If $\tau(\varphi_{n}(1_{k}))>1-(\sigma/2)^{2},$ then, by (e 4.45),

$\displaystyle\tau(\varphi_{n+1}(1_{k}))>1-(\varepsilon/2^{n+7})^{2}-\sigma/2-(\varepsilon/2^{2(n+1+2)})^{2}>1-(\varepsilon/2^{n+1+4})^{2}-\sigma/2.$
(e 4.52)

This completes the induction and the lemma follows. ∎

##  5 The main results

Let us first recall a result of W. Zhang:

######  Lemma 5.1 (W. Zhang, Lemma 6.5 of [61]).

Let $d\geq 0$ and $k\geq 2$ be integers and let $A$ be a non-elementary
separable simple unital amenable $C^{*}$-algebra with $T(A)\not=\emptyset.$
Then, for any compact subset $Y\subset\partial_{e}(T(A))$ with ${\rm
dim}(Y)\leq d,$ any finite subset ${\cal F}\subset A$ and $\varepsilon>0,$
there exists an order zero c.p.c. map $\varphi:M_{k}\to A$ such that

$\displaystyle\|[\varphi(b),\,y]\|<\varepsilon\|b\|\,\,\,\text{for\,\,\,all}\,\,\,b\in
M_{k},\,\,y\in{\cal F}\,\,\,\text{and}\,\,\,$ (e 5.1)
$\displaystyle\tau(\varphi(1_{k}))>1-\varepsilon\,\,\,\text{for\,\,\,all}\,\,\,\tau\in
Y.$ (e 5.2)

###### Proof.

Since the proof of this is not provided in [61], let us explain briefly. As
pointed out in [61], the proof only depends on Lemma 6.4 of [61] (whose proof
is also omitted in [61]) which is referred to Lemma 3.5 of [58]. Note that the
finite open cover argument remains exactly the same as that of the proof of
Lemma 3.5 of [58] (replacing $\partial_{e}(T(A))$ by compact subset $Y,$ and
open subsets $V_{j}^{(i)}$ by relatively open subsets of $Y$). One point
should be made clearer is the extension of functions $f_{j}^{(i)}$ (part of
the partition of unity) in the proof of Lemma 3.5 of [58]. Indeed, one only
defines $f_{j}^{(i)}$ on the compact subset $Y.$ Applying Corollary 11.15 of
[31], one may extend $f_{j}^{(i)}$ to real affine continuous functions on
$T(A).$ So the rest of the proof of Lemma 3.5 of [58] remains the same (but
restricting on $Y$ only). In other words, Lemma 6.4 of [61] follows.
Consequently, as pointed out in [61], Lemma 6.5 of [61] holds. ∎

######  Remark 5.2.

Lemma 5.1 may also be extracted from [52] and [35]. In [58] as well as in
[35], the condition that $T(A)$ is a Bauer simplex is required. It is worth
noting that Ken Goodearl’s Corollary 11.15 of [31] plays a key role in the
statements in [61] which allows Wei Zhang to work on a compact subset of
$\partial_{e}(T(A))$ instead of $\partial_{e}(T(A)).$ Next we will explain
that W. Zhang’s lemma also holds without assuming $A$ is unital.

Let us state the following lemma which is known in the unital case.

######  Lemma 5.3 (Lemma 4.2 of [52], Lemma 3.3 and 3.4 of [56]).

Let $A$ be a separable simple amenable $C^{*}$-algebra with nonempty compact
$T(A)$ and $F\subset\partial_{e}(T(A))$ be a compact subset. For any mutually
orthogonal positive functions $f_{1},f_{2},...,f_{m}$ in $C(F),$ there exist
$\\{a_{i,n}\\}_{n\in\mathbb{N}}\in A^{\prime},$ $1\leq i\leq m,$ such that

$\displaystyle\lim_{n\to\infty}\max_{\tau\in
F}|\tau(a_{i,n})-f_{i}(\tau)|=0\,\,\,{\rm
and}\,\,\,\lim_{n\to\infty}\|a_{i,n}a_{j,n}\|=0,\,\,i\not=j,\,\,1\leq j\leq
m.$ (e 5.3)

###### Proof.

First one observes that Lemma 3.3 and Lemma 3.4 in [56] hold for the case that
$A$ is not necessarily unital but $T(A)$ is compact (see also the footnote on
the page with Lemma 3.6, also Lemma 4.1 of [52]). Then, by Theorem 11.14 of
[31], there are $g_{1},g_{2},...,g_{m}\in\operatorname{Aff}(T(A))$ with $0\leq
g_{i}\leq 1$ such that $(g_{i})|_{F}=f_{i},$ $1\leq i\leq m.$ By (non-unital
version of) Lemma 3.3 of [56] mentioned, one obtains $\\{e_{n}^{(1)}\\}\in
A^{\prime},$ such that

$\displaystyle\lim_{n\to\infty}\max_{\tau\in
T(A)}|\tau(e_{n}^{(1)})-g_{1}(\tau)|=0.$ (e 5.4)

Then, by Proposition 3.1 of [11] and (non-unital version of) Lemma 3.3 of
[56], one obtains $\\{e_{n}^{(2)}\\}\in A^{\prime},$ such that

$\displaystyle\lim_{n\to\infty}\max_{\tau\in
T(A)}|\tau(e_{n}^{(2)})-g_{2}(\tau)|=0\,\,\,{\rm
and}\,\,\,\lim_{n\to\infty}\max_{\tau\in F}|\tau(e^{(1)}_{n}e^{(2)}_{n})|=0.$
(e 5.5)

Repeating this, one obtains $\\{e_{n}^{(i)}\\}\in A^{\prime},$ $1\leq i\leq
m,$ such that

$\displaystyle\lim_{n\to\infty}\max_{\tau\in
T(A)}|\tau(e_{n}^{(i)})-g_{i}(\tau)|=0\,\,\,{\rm
and}\,\,\,\lim_{n\to\infty}\max_{\tau\in
F}|\tau(e^{(i)}_{n}e^{(j)}_{n})|=0,\,\,i\not=j.$ (e 5.6)

The lemma then follows by applying (non-unital version of) Lemma 3.4 of in
[56] mentioned above. ∎

We would like to state the non-unital version of W. Zhang’s Lemma 5.1 as
follows.

######  Lemma 5.4 (Proposition 5.1 of [52] and Lemma 6.5 of [61]).

Let $A$ be a non-elementary separable amenable simple $C^{*}$-algebra with
compact $T(A)\not=\emptyset$ and let $S\subset\partial_{e}(T(A))$ be a compact
subset with covering dimension $d\in\mathbb{Z}_{+}.$ Then, for any
$k\in\mathbb{N},$ there exists a unital homomorphism
$\varphi:M_{k}\to(l^{\infty}(A)\cap A^{\prime})/I_{{}_{S,\varpi}}.$

###### Proof.

This is actually proved in Proposition 5.1 of [52]. We will use the second
part of Proposition 3.1 of [11] and Lemma 5.3 to replace Lemma 4.2 of [52]. It
suffices to show that the conclusion of Proposition 5.1 of [52] holds but
replacing $A_{t,\infty}$ (the notation used in Proposition 5.1 of [52]) by
$(l^{\infty}(A)\cap A^{\prime})/_{I_{S,\mathbb{N}}}$ (choosing
$\varphi:=\sum_{l=1}^{d}\varphi_{l}$). In other words, it suffices to show
that there exist $\varphi_{l,n}:M_{k}\to A$ which satisfies (i)’ (replacing
$1_{A_{\infty}}$ by $1_{l^{\infty}(A)/I_{{}_{S,\mathbb{N}}}}$), (ii)’, (iii)’,
(iv)’ and (v)’ (stated in the proof of [52]) hold (for $B=S$).

Now as in the proof of Proposition 5.1, let $B\subset\partial_{e}(T(A))$ be a
compact subset with covering dimension $c\leq d.$ As in the proof of
Proposition 5.1 of [52], using induction on $d,$ it suffices to prove that
there exists a c.p.c. map $\varphi_{l,n}:M_{k}\to A,$ $l=0,1,2,...,c\,(\leq
d),$ $n\in\mathbb{N}$ such that (i)’, (ii)’, (iii)’, (iv)’ and (v)’ (stated in
the proof of [52]) hold. In fact the proof of Proposition 5.1 of [52] does
just that (with one obvious modification: (a-vii) holds when we replace
$\partial_{e}(T(A))$ by $B$ as we apply the second part of Proposition 3.1 of
[11] instead of part (i) of Lemma 4.2 of [52] and Lemma 5.3 instead of part
(ii) of Lemma 4.2 of [52]). ∎

######  Lemma 5.5.

Let $A$ be a separable non-elementary amenable simple $C^{*}$-algebra with
$T(A)\not=\emptyset$ which is compact and with T-tracial approximate
oscillation zero such that $\partial_{e}(T(A))$ satisfying condition (C).
Then, for any integer $k\in\mathbb{N},$ any $\varepsilon>0$ and any finite
subset ${\cal F}\subset A,$ there is an order zero c.p.c. map
$\varphi:M_{k}\to A$ such that

$\displaystyle\tau(\varphi(1_{k}))>1-\varepsilon\,\,\,\text{for\,\,\,all}\,\,\,\tau\in
T(A)\,\,\,\text{and}\,\,\,\|[f,\,\varphi(b)]\|<\varepsilon$ (e 5.7)

for all $f\in{\cal F}$ and $b\in M_{k}^{\bf 1}.$

###### Proof.

Since $T(A)$ satisfies condition (C), we may write
$\partial_{e}(T(A))=\cup_{n=1}^{\infty}X_{n}$ which satisfies conditions (1)
and (2) in 2.12. Write $S_{n}=\cup_{i=1}^{n}X_{i}.$ Note that each $S_{n}$ has
finite covering dimension and compact. By Lemma 5.4, there exists, for each
$n\in\mathbb{N},$ a unital homomorphism
${\bar{\psi}}:M_{k}\to(l^{\infty}(A)\cap A^{\prime})/I_{{}_{S_{n},\varpi}}.$
Since $A$ is assumed to have T-tracial approximate oscillation zero, by Lemma
3.9, there exists, for each $n\in\mathbb{N},$ a homomorphism
$\psi:M_{k}\to(l^{\infty}(A)\cap A^{\prime})/I_{{}_{T(A),\varpi}}$ such that
$\Phi_{S_{n}}\circ\psi={\bar{\psi}},$ where
$\Phi_{S_{n}}:l^{\infty}(A)/_{{}_{T(A),\varpi}}\to
l^{\infty}(A)/I_{{}_{S_{n},\varpi}}$ is the quotient map, and $\psi$ has
property (Os).

Fix $k\in\mathbb{N},$ $\varepsilon\in(0,1/2)$ and a finite subset ${\cal
F}\subset A.$

Applying Lemma 4.2, we obtain a sequence of order zero c.p.c. maps
$\varphi_{n}:M_{k}\to A$ which satisfies the conclusion of Lemma 4.2 (with
respective to $\\{S_{n}\\},$ ${\cal F}$ and $\varepsilon$).

Put $G_{n}=\\{\tau\in T(A):\tau(\varphi_{n}(1_{k}))>1-(\varepsilon/4)^{2}\\},$
$n\in\mathbb{N}.$

Let $\tau\in T(A).$ Then, by The Choquet Theorem, there is a probability Borel
measure $\mu_{\tau}$ on $\partial_{e}(T(A))$ such that

$\displaystyle\tau(f)=\int_{\partial_{e}(T(A))}fd\mu_{\tau}\,\,\,{\rm
for\,\,\,all}\,\,\,f\in\operatorname{Aff}(T(A)).$ (e 5.8)

Let $\mu_{\tau,n}=\mu_{\tau}|_{X_{n}}.$ Then

$\displaystyle\tau(f)=\sum_{n=1}^{\infty}\int_{X_{n}}fd\mu_{\tau,n}\,\,\,{\rm
for\,\,\,all}\,\,\,f\in\operatorname{Aff}(T(A)).$ (e 5.9)

Since $\|\mu_{\tau}\|=1,$ there is $n_{1}\in\mathbb{N}$ such that

$\displaystyle\sum_{m>n_{1}}^{\infty}\|\mu_{\tau,m}\|<(\varepsilon/8)^{2}\,\,\,{\rm
and}\,\,\,\mu_{\tau}(S_{n_{1}})>1-(\varepsilon/8)^{2}.$ (e 5.10)

We may assume that $n_{1}>2.$ Then (as $\\{\varphi_{n}\\}$ satisfies the
conclusion of Lemma 4.2), if $n\geq n_{1},$

$\displaystyle\tau(\varphi_{n}(1_{k}))$ $\displaystyle=$
$\displaystyle\int_{S_{n_{1}}}\widehat{\varphi_{n}(1_{k})}(s)d\mu_{\tau}+\sum_{m>n_{1}}\int_{F_{m}}\widehat{\varphi_{n}(1_{k})}(s)d\mu_{\tau,m}$
(e 5.11) $\displaystyle\geq$
$\displaystyle(1-(\varepsilon/2^{n+5})^{2})\mu_{\tau}(S_{n_{1}})>1-(\varepsilon/4)^{2}.$
(e 5.12)

In other words, $\tau\in G_{n}$ (for $n\geq n_{1}$). It follows that
$\cup_{n=1}^{\infty}G_{n}\supset T(A).$ Since $T(A)$ is compact, there exists
$n_{0}\in\mathbb{N}$ such that

$\displaystyle T(A)\subset\cup_{n=1}^{n_{0}}G_{n}.$ (e 5.13)

Let $\tau\in T(A).$ Suppose that $\tau\in G_{j}$ for some $j\leq n_{0}.$ Then

$\displaystyle\tau(\varphi_{j}(1_{k}))>1-(\varepsilon/4)^{2}.$ (e 5.14)

It follows from the conclusion of Lemma 4.2 that

$\displaystyle\tau(\varphi_{n_{0}}(1_{k}))>1-\sum_{i=j+1}^{n_{0}}\varepsilon/2^{i+1+4}-\varepsilon/4>1-\varepsilon.$
(e 5.15)

Choose $\varphi=\varphi_{n_{0}}.$ Then

$\displaystyle\|[x,\,\varphi(b)]\|<\varepsilon\,\,\,{\rm
for\,\,\,all}\,\,\,x\in{\cal F}\,\,\,{\rm and}\,\,\,b\in M_{k}^{\bf
1},\,\,\,{\rm and}\,\,\,\tau(\varphi(1_{k}))>1-\varepsilon\,\,\,{\rm
for\,\,\,all}\,\,\,\tau\in T(A).$ (e 5.16)

The lemma follows.

∎

######  Theorem 5.6.

Let $A$ be a separable non-elementary amenable simple $C^{*}$-algebra whose
extremal boundaries of $\widetilde{T}(A)\setminus\\{0\\}$ satisfies condition
(C) (see Definition 2.18). Then following are equivalent.

(1) $A$ has strict comparison and T-tracial approximate oscillation zero,

(2) $A$ has strict comparison and stable rank one,

(3) $A\cong A\otimes{\cal Z},$

(4) $A$ has finite nuclear dimension.

###### Proof.

We only need to prove (1) $\Rightarrow$ (3). The equivalence of (1) and (2)
follows from Theorem 1.1 of [26]. The equivalence of (3) and (4) has been
proved (see [10], [9], [59], [54], [44]) in general (without the assumption
that $T(A)$ satisfies condition (C)). That (3) implies that $A$ has strict
comparison is proved in [50] in general. In fact, it is also proved that, in
unital case, $A$ has stable rank one. For non-unital case, that $A$ is ${\cal
Z}$-stable implies that $A$ has stable rank one is proved in [25] (Corollary
6.8 of [25]). In other words, (3) $\Rightarrow$ (2) holds (without the
assumption that $T(A)$ satisfies condition(C)).

To show (1) $\Rightarrow$ (3), let us assume that $A$ has strict comparison
and T-tracial approximate oscillation zero. It follows from Theorem 7.12 of
[26] that the canonical map $\Gamma:\mathrm{Cu}(A)\to{\rm
LAff}_{+}({\widetilde{T}}(A))$ is surjective. Then, choose $a\in{\rm
Ped}(A)_{+}^{\bf 1}\setminus\\{0\\}$ such that $d_{\tau}(a)$ is continuous on
${\widetilde{T}}(A).$ Define $A_{1}=\mathrm{Her}(a).$ Then $A_{1}$ has
continuous scale and $T(A_{1})$ is compact (see Proposition 5.4 of [22]).
Since $A_{1}\otimes{\cal K}\cong A\otimes{\cal K}$ ([6]), by Cor. 3.1 of [57],
it suffices to show that $A_{1}$ is ${\cal Z}$-stable. It follows part (2) of
Proposition 2.17 that $T(A_{1})$ satisfies condition (C).

Fix an integer $k\geq 2.$ Let $\\{{\cal F}_{n}\\}$ be an increasing sequence
of finite subsets of $A_{1}$ such that $\cup_{n=1}^{\infty}{\cal F}_{n}$ is
dense in $A_{1}.$ By Lemma 5.5, for each $n\in\mathbb{N},$ there exists an
order zero c.p.c. map $\varphi_{n}:M_{k}\to A_{1}$ such that

$\displaystyle\|[a,\,\varphi_{n}(b)]\|<1/n\,\,\,{\rm
for\,\,\,all}\,\,\,a\in{\cal F}_{n}\,\,\,{\rm and}\,\,\,b\in M_{k}^{\bf
1}\,\,\,{\rm and}\,\,\,$ (e 5.17)
$\displaystyle\sup\\{\tau(\varphi_{n}(1_{k})):\tau\in
T(A_{1})\\}>1-1/n,\,\,\,n\in\mathbb{N}.$ (e 5.18)

Define $\Phi:M_{k}\to l^{\infty}(A_{1})$ by $\Phi(b)=\\{\varphi_{n}(b)\\}.$
Then, by (e 5.17), $\Phi$ maps $M_{k}$ into
$l^{\infty}(A_{1})\cap(A_{1})^{\prime}.$ By (e 5.18),

$\displaystyle\lim_{n\to\infty}\sup\\{1-\tau(\varphi_{n}(1_{k})):\tau\in
T(A_{1})\\}=0.$ (e 5.19)

It follows that $\Pi_{\varpi}\circ\Phi$ is a unital order zero c.p.c. map.
Therefore it is a unital homomorphism. Hence (2) follows from a result of
Matui-Sato (see, explicitly, Corollary 5.11 and Proposition 5.12 of [35], for
example) in the unital case. For non-unital case, we take a detour and use the
result in [12]. In this case, since $T(A_{1})$ is compact, $A_{1}$ is
uniformly McDuff (see Definition 4.1 of [12], or Definition 4.2 of [11]).
Since $A_{1}$ has strict comparison, by Proposition 4.4 of [12] (also a
version of Matui-Sato’s result), we conclude that $A_{1}\cong
A_{1}\otimes{\cal Z}.$ ∎

######  Corollary 5.7.

Let $A$ be a separable non-elementary amenable simple $C^{*}$-algebra with
${\widetilde{T}}(A)\not=\emptyset$ which has a basis $S$ satisfying condition
(C) and $\partial_{e}(S)$ has countably many points. Then following are
equivalent.

(1) $A$ has strict comparison,

(2) $A\cong A\otimes{\cal Z},$

(3) $A$ has finite nuclear dimension.

###### Proof.

We note, by Theorem 1.1 of [26], $A$ has T-tracial approximate oscillation
zero.

∎

######  Remark 5.8.

In Theorem 5.6, by (2) of Proposition 2.17, the condition on
$\widetilde{T}(A)$ does not depend on the choice of the basis $S.$ If $A$ is
unital, or $A$ has continuous scale, the condition can be stated as $T(A)$
satisfies condition (C) (in particular, $T(A)$ may not be a Bauer simplex and
may not have finite covering dimension). One day after the first version of
this paper was posted, we were informed by Kang Li the paper of Wei Zhang [61]
which contains statements overlapping with our original statement. This helps
us to improve our main results. It is, perhaps, conceivable, that the
condition that a basis $S$ of $\widetilde{T}(A)$ satisfies condition (C) can
be further weakened in Theorem 5.6.

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