# comparison properties of asymptotically tracially approximation ${\rm
C^{*}}$-algebras

Qingzhai Fan and Xiaochun Fang  Qingzhai Fan
Department of Mathematics
Shanghai Maritime University
Shanghai
China
201306<EMAIL_ADDRESS>Xiaochun Fang
Department of Mathematics
Tongji University
Shanghai
China
200092<EMAIL_ADDRESS>
###### Abstract.

We show that the following properties of the ${\rm C^{*}}$-algebras in a class
$\mathcal{P}$ are inherited by simple unital ${\rm C^{*}}$-algebras in the
class of asymptotically tracially in $\mathcal{P}$: $(1)$ $\beta$-comparison
(in the sense of Kirchberg and Rørdam), $(2)$ $n$-comparison (in the sense of
Winter).

Key words ${\rm C^{*}}$-algebras, asymptotically tracially approximation,
Cuntz semigroup.

2000 _Mathematics Subject Classification._ 46L35, 46L05, 46L80

##  1\. Introduction

Tracial topological rank no more than $k$ was introduced by Lin in [18].
Instead of assuming inductive limit structure, he started with a certain
abstract tracial approximation property, and ${\rm C^{*}}$-algebras with
tracial topological rank no more than one and certain additional properties
are AH algebras without dimension growth which was classified by Elliott-Gong
in [3] (in the real rank zero case) and classified by Elliott-Gong-Li in [4].

This abstract tracial approximation structure has proved to be very important
in the classification of simple amenable ${\rm C^{*}}$-algebras. For example,
it led to the classification of unital simple separable amenable ${\rm
C^{*}}$-algebras with finite nuclear dimension in the UCT class (see [12],
[5], [28]).

Examples of Rørdam [26] and Toms [29], relying on techniques pioneered by
Villadsen [31], demonstrated that some sort of regularity condition, stronger
than nuclearity, is necessary in order to have a classification by $K$-theory
and trace. Three regularity conditions have emerged: finite nuclear dimension,
tensorial absorption of the Jiang-Su algebra $\mathcal{Z},$ and algebraic
regularity in the Cuntz semigroup. Toms and Winter have conjectured (see e.g.
[7]) that these three fundamental properties are equivalent for all separable,
simple, unital, amenable ${\rm C^{*}}$-algebras.

Kirchberg and Rørdam introduced a weaker comparison property and also a
property of a ${\rm C^{*}}$-algebra called $\beta$-comparison in [16]. The
property of $n$-comparison was introduced by Winter in [32].

It is an open problem if Kirchberg’s and Rørdam’s weak comparison and
$\beta$-comparison, Winter’s $m$-comparison, and strict comparison all agree
for simple unital ${\rm C^{*}}$-algebras. (Somewhat confusingly, it is known
that $m$-comparison for a particular $m$ does agree with $\beta$-comparison
for $\beta=m+1$.)

In order to search a tracial version of Toms-Winter conjecture, also inspire
by tracial $\mathcal{Z}$-absorbing ${\rm C^{*}}$-algebras which was introduced
by Hirshberg and Orovitz in [15], Fu and Lin introduced a class of
asymptotically tracially approximation ${\rm C^{*}}$-algebras in [10].

The following definition is not exactly same as definition 3.1 in [10], but by
proposition 3.8 in [10], the following definition is equivalent to the
definition 3.1 in [10].

Let $\mathcal{P}$ be a class of unital ${\rm C^{*}}$-algebras, the class of
simple unital ${\rm C^{*}}$-algebras which can be asymptotically tracially in
$\mathcal{P}$, denoted by ${\rm ATA}\mathcal{P}$.

######  Definition 1.1.

([10].) A simple unital ${\rm C^{*}}$-algebra $A$ is said to belong to the
class ${\rm ATA}\mathcal{P}$ if, for any $\varepsilon>0,$ any finite subset
$F\subseteq A,$ and any non-zero element $a\geq 0,$ there exist a ${\rm
C^{*}}$-algebra $B$ in $\mathcal{P}$ and completely positive contractive
linear maps $\alpha:A\to B$ and $\beta_{n}:B\to A$, and $\gamma_{n}:A\to
A\cap\beta_{n}(B)^{\perp}$ such that

$(1)$ the map $\alpha$ is unital completely positive linear map,
$\beta_{n}(1_{B})$ and $\gamma_{n}(1_{A})$ are projections and
$\beta_{n}(1_{B})+\gamma_{n}(1_{A})=1_{A}$ for all $n\in\mathbb{N}$,

$(2)$ $\|x-\gamma_{n}(x)-\beta_{n}(\alpha(x))\|<\varepsilon$ for all $x\in F$
and for all $n\in{\mathbb{N}}$,

$(3)$ $\alpha$ is an $F$-$\varepsilon$ approximate embedding,

$(4)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(xy)\|=\|x\|$ for all $x,y\in B$, and

$(5)$ $\gamma_{n}(1)\lesssim a$ for all $n\in\mathbb{N}$.

In [10], Fu and Lin show that the following properties of unital ${\rm
C^{*}}$-algebras in a class $\mathcal{P}$ are inherited by simple unital ${\rm
C^{*}}$-algebras in the class ${\rm ATA}\mathcal{P}$: $(1)$ stably finite,
$(2)$ quasidiagonal ${\rm C^{*}}$-algebras, $(3)$ purely infinite simple ${\rm
C^{*}}$-algebras, $(4)$ tracial $\mathcal{Z}$-absorption, $(5)$ the Cuntz
semigroup is almost unperforated, and $(6)$ strict comparison property.

In this paper, we show that the following comparison properties of unital
${\rm C^{*}}$-algebras in a class $\mathcal{P}$ are inherited by simple unital
${\rm C^{*}}$-algebras in the class of asymptotically tracially in
$\mathcal{P}$: $(1)$ $\beta$-comparison ( in the sense of Kirchberg and
Rørdam; see [16]), $(2)$ $n$-comparison (in the sense of Winter; see [32]).

##  2\. Definitions and preliminaries

Let ${\rm M}_{n}(A)_{+}$ denote the positive elements of ${\rm M}_{n}(A)$.
Given $a,b\in{\rm M}_{n}(A)_{+},$ we say that $a$ is Cuntz subequivalent to
$b$ (written $a\lesssim b$) if there is a sequence $(v_{n})_{n=1}^{\infty}$ of
elements of ${\rm M}_{n}(A)$ such that

$\lim_{n\to\infty}\|v_{n}bv_{n}^{*}-a\|=0.$

We say that $a$ and $b$ are Cuntz equivalent (written $a\sim b$) if $a\lesssim
b$ and $b\lesssim a$. We write $\langle a\rangle$ for the equivalence class of
$a$.

The object ${\rm W}(A):={\rm M}_{\infty}(A)_{+}/\sim$ will be called the Cuntz
semigroup of $A$. (See [2].) ${\rm W}(A)$ becomes an ordered semigroup when
equipped with the addition operation

$\langle a\rangle+\langle b\rangle=\langle a\oplus b\rangle,$

and the order relation

$\langle a\rangle\leq\langle b\rangle\Leftrightarrow a\lesssim b.$

Given $a$ in ${\rm M}_{\infty}(A)_{+}$ and $\varepsilon>0,$ we denote by
$(a-\varepsilon)_{+}$ the element of ${\rm C^{*}}(a)$ corresponding (via the
functional calculus) to the function
$f(t)={\max(0,t-\varepsilon)},~{}t\in\sigma(a)$.

We shall say that a separable exact ${\rm C^{*}}$-algebra $A$ has strict
comparison if for $a,~{}b\in{\rm M}_{k}(A)_{+},$ with
$d_{\tau}(a)<d_{\tau}(b)$ for any $\tau\in{\rm T}(A),$ then we have $a\lesssim
b$, where ${\rm T}(A)$ is the set of tracial states of $A$.

######  Definition 2.1.

([32]) Let $A$ be a unital ${\rm C^{*}}$-algebra. We say $A$ has
$n$-comparison, if, whenever $x,~{}y_{0},~{}y_{1},$ $~{}y_{2},$
$\cdots,~{}y_{n}$ are elements in ${\rm W}(A)$ such that $x<_{s}y_{j}$ for all
$j=0,~{}1,~{}\cdots,~{}n,$ then $x\leq y_{0}+y_{1}+\cdots+y_{n}$. Here,
$x<_{s}y$ means $(k+1)x\leq ky$ for some natural number $k$.

######  Definition 2.2.

([16]) Let $A$ be a unital ${\rm C^{*}}$-algebra and let $1\leq\beta<\infty$.
We say that $A$ has $\beta$-comparison if for all $x,y\in$ ${\rm W}(A)$ and
all integers $k,~{}l\geq 1$ with $k>\beta l,$ the inequality $kx\leq ly$
implies $x\leq y$.

It follows immediately from the definitions that ${\rm W}(A)$ is almost
unperforated if and only if $A$ has 1-comparison in the sense of Kirchberg and
Rørdam.

######  Theorem 2.3.

([1], [15].) Let $A$ be a stably finite ${\rm C^{*}}$-algebra.

$(1)$ Let $a,~{}b\in A_{+}$ and $\delta>0$ be such that $\|a-b\|<\delta$. Then
we have $(a-\delta)_{+}\lesssim b$.

$(2)$ Let $a$ be a purely positive element of $A$ (i.e., $a$ is not Cuntz
equivalent to a projection). Let $\eta>0$, and let $f\in C_{0}(0,1]$ be a non-
negative function with $f=0$ on $(\eta,1),$ $f>0$ on $(0,\eta)$, and
$\|f\|=1$. We have $f(a)\neq 0$ and $(a-\eta)_{+}+f(a)\lesssim a.$

$(3)$ The following conditions are equivalent: $(1)^{\prime}$ $a\lesssim b,$
$(2)^{\prime}$ for any $\delta>0,$ $(a-\delta)_{+}\lesssim b,$ and
$(3)^{\prime}$ for any $\delta>0$, there is $\varepsilon>0,$ such that
$(a-\delta)_{+}\lesssim(b-\varepsilon)_{+}.$

######  Lemma 2.4.

([10].) If the class $\mathcal{P}$ is closed under tensoring with matrix
algebras and under passing to unital hereditary ${\rm C^{*}}$-subalgebras,
then the class ${\rm ATA}\mathcal{P}$ is closed under tensoring with matrix
algebras and under passing to unital hereditary ${\rm C^{*}}$-subalgebras.

The following lemma is obvious, and we omit the proof.

######  Lemma 2.5.

The $n$-comparison (or $\beta$-comparison) is preserved under tensoring with
matrix algebras and under passing to unital hereditary ${\rm
C^{*}}$-subalgebras.

##  3\. The main results

######  Theorem 3.1.

Let $\mathcal{P}$ be a class of stably finite unital ${\rm C^{*}}$-algebras
which have $\beta$-comparison (in the sense of Kirchberg and Rørdam), for some
$1\leq\beta<\infty$. Then $A$ has $\beta$-comparison for any simple unital
${\rm C^{*}}$-algebra $A\in{\rm ATA}\mathcal{P}.$

###### Proof.

By Lemma 2.5 and Lemma 2.4, enlarging the class $\mathcal{P}$, we may suppose
it is closed under passing to matrix algebras and unital hereditary ${\rm
C^{*}}$-subalgebras (i.e., Morita equivalent ${\rm C^{*}}$-algebras).

Let $a,b\in{\rm M}_{\infty}(A)_{+}$. By Theorem 2.3 (3), we need to show that
$\langle(a-\varepsilon)_{+}\rangle\leq\langle b\rangle$ for any
$\varepsilon>0,$ and any integers $k,~{}l\geq 1$ such that $k>\beta l,$
$k\langle a\rangle\leq l\langle b\rangle.$

We may assume that $a,b\in{\rm M}_{n}(A)_{+}$ for some integer $n$. By Lemma
2.4 and Lemma 2.5, we may assume that $a,b\in A_{+}$ and $\|a\|\leq
1,\|b\|\leq 1$.

We divide the proof into three cases.

$(\textbf{I})$, we suppose that $b$ is not Cuntz equivalent to a projection.

Given $\delta>0$, and $k,l\geq 1$ as above, with in particular $k>l$, since
$k\langle a\rangle\leq l\langle b\rangle.$ Hence, by Theorem 2.3 (1), there
exists $v=(v_{i,j})\in{\rm M}_{k}(A),~{}1\leq i\leq k,~{}1\leq j\leq k$ such
that

$\|v({\rm diag}(b\otimes 1_{l},~{}0\otimes 1_{k-l})){v}^{*}-a\otimes
1_{k}\|<\delta.$

By Theorem 2.3 (2), there is a non-zero positive element $d$ orthogonal to $b$
such that

$(b-\delta/2)_{+}+d\lesssim b.$

With $F=\\{a,~{}b,~{}v_{i,j}:~{}1\leq i\leq k,~{}1\leq j\leq k\\},$ and
$\varepsilon^{\prime}>0$, since $A\in{\rm ATA}\mathcal{P},$ there exist a
${\rm C^{*}}$-algebra $B$ in $\mathcal{P}$ and completely positive contractive
linear maps $\alpha:A\to B$ and $\beta_{n}:B\to A$, and $\gamma_{n}:A\to
A\cap\beta_{n}(B)^{\perp}$ such that

$(1)$ the map $\alpha$ is unital completely positive linear map,
$\beta_{n}(1_{B})$ and $\gamma_{n}(1_{A})$ are all projections, and
$\beta_{n}(1_{B})+\gamma_{n}(1_{A})=1_{A}$ for all $n\in\mathbb{N}$,

$(2)$ $\|x-\gamma_{n}(x)-\beta_{n}(\alpha(x))\|<\varepsilon^{\prime}$ for all
$x\in F$ and for all $n\in{\mathbb{N}}$,

$(3)$ $\alpha$ is an $F$-$\varepsilon^{\prime}$ approximate embedding,

$(4)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(xy)\|=\|x\|$ for all $x,y\in B$, and

$(5)$ $\gamma_{n}(1)\lesssim d$ for all $n\in\mathbb{N}$.

Since $\|v{\rm diag}(b\otimes 1_{l},~{}0\otimes 1_{k-l}){v}^{*}-a\otimes
1_{k}\|<\delta$, by $(1)$, we have

$\|\alpha\otimes id_{M_{k}}(v({\rm diag}(b\otimes 1_{l},~{}0\otimes
1_{k-l})){v}^{*})-\alpha(a)\otimes 1_{k}\|<\delta.$

By $(3)$, we have

$\|\alpha\otimes id_{M_{k}}(v)({\rm diag}(\alpha(b)\otimes 1_{l},~{}0\otimes
1_{k-l}))\alpha\otimes id_{M_{k}}({v}^{*})-\alpha(a)\otimes 1_{k}\|<2\delta.$

By Theorem 2.3 (1), we have

$k\langle(\alpha(a)-2\delta)_{+}\rangle\leq l\langle\alpha(b)\rangle.$

Since $B\in\Omega$, we have

$\langle(\alpha(a)-2\delta)_{+}\rangle\leq\langle\alpha(b)\rangle.$

Since $\langle(\alpha(a)-2\delta)_{+}\rangle\leq\langle\alpha(b)\rangle$,
there exist $w\in B$ such that

$\|w\alpha(b)w^{*}-(\alpha(a)-2\delta)_{+}\|<\delta.$

Since $\|w\alpha(b)w^{*}-(\alpha(a)-2\delta)_{+}\|<\delta,$ we have

$\|\beta_{n}(w\alpha(b)w^{*})-\beta_{n}((\alpha(a)-2\delta)_{+})\|<\delta.$

By $(4)$ we have

$\|\beta_{n}(w)\beta_{n}(\alpha(b))\beta_{n}(w^{*})-\beta_{n}((\alpha(a)-2\delta)_{+})\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle(\beta_{n}((\alpha(a)-6\delta)_{+})-2\delta)_{+}\rangle\leq\langle(\beta_{n}\alpha(b)-2\delta)_{+}\rangle.$

Therefore, we have

$\displaystyle\langle(a-\varepsilon)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}(a)\rangle+\langle(\beta_{n}\alpha(a)-20\delta)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}(1_{A})\rangle+\langle(\beta_{n}\alpha(a)-20\delta)_{+})\rangle$
$\displaystyle\leq\langle
d\rangle+\langle((\beta_{n}(\alpha(a)-2\delta)_{+})-6\delta)_{+}\rangle$
$\displaystyle\leq\langle(b-\delta/2)_{+}\rangle+\langle d\rangle\leq\langle
b\rangle.$

$(\textbf{II})$, we suppose that $b$ is Cuntz equivalent to a projection and
$a$ is not Cuntz equivalent to a projection. Choose a projection $p_{0}$ such
that $b$ is Cuntz equivalent to $p_{0}.$ We may assume that $b=p_{0}.$

By Theorem 2.3 (2), let $\delta>0$ (with $\delta<\varepsilon$), there exists a
non-zero positive element $d$ orthogonal to $a$ such that
$\langle(a-\delta)_{+}+d\rangle\leq\langle a\rangle.$ Since $k\langle
a\rangle\leq l\langle p_{0}\rangle,$ we have
$k\langle(a-\delta)_{+}+d\rangle\leq l\langle p_{0}\rangle.$ Hence, as above,
by Theorem 2.3 (1), on increasing $\delta$ slightly, and decreasing $d$
slightly, there exists $v=(v_{i,j})\in{\rm M}_{k}(A),~{}1\leq i\leq k,~{}1\leq
j\leq k$, such that

$\|v{\rm diag}(p_{0}\otimes 1_{l},~{}0\otimes
1_{k-l})v^{*}-((a-\delta)_{+}+d)\otimes 1_{k}\|<\delta.$

Since $A\in{\rm ATA}\mathcal{P},$ for
$F=\\{a,~{}p_{0},~{}d,~{}{v_{i,j}}:~{}1\leq i\leq k,~{}1\leq j\leq k\\},$ and
$\varepsilon^{\prime}>0$, there exist a ${\rm C^{*}}$-algebra $B$ in
$\mathcal{P}$ and completely positive contractive linear maps $\alpha:A\to B$
and $\beta_{n}:B\to A$, and $\gamma_{n}:A\to A\cap\beta_{n}(B)^{\perp}$ such
that

$(1)$ the map $\alpha$ is unital completely positive linear map,
$\beta_{n}(1_{B})$ and $\gamma_{n}(1_{A})$ are all projections,
$\beta_{n}(1_{B})+\gamma_{n}(1_{A})=1_{A}$ for all $n\in\mathbb{N}$,

$(2)$ $\|x-\gamma_{n}(x)-\beta_{n}(\alpha(x))\|<\varepsilon^{\prime}$ for all
$x\in F$ and for all $n\in{\mathbb{N}}$.

$(3)$ $\alpha$ is an $F-\varepsilon^{\prime}$-approximate embedding,

$(4)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(xy)\|=\|x\|$ for all $x,y\in B$.

Since $\|v{\rm diag}(p_{0}\otimes 1_{l},~{}0\otimes
1_{k-l})v^{*}-((a-\delta)_{+}+d)\otimes 1_{k}\|<\delta$, by $(1)$, we have

$\|\alpha\otimes id_{M_{k}}(v({\rm diag}(p_{0}\otimes 1_{l},~{}0\otimes
1_{k-l})){v}^{*})-\alpha((a-\delta)_{+}+d)\otimes 1_{k}\|<\delta.$

By $(3)$, we have

$\|\alpha\otimes id_{M_{k}}(v)({\rm diag}(\alpha(p_{0})\otimes
1_{l},~{}0\otimes 1_{k-l}))\alpha\otimes
id_{M_{k}}({v}^{*})-\alpha((a-\delta)_{+}+d)\otimes 1_{k}\|<2\delta.$

By Theorem 2.3 (1), we have

$k\langle(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\rangle\leq
l\langle\alpha(b)\rangle.$

Since $B\in\Omega$, we have

$\langle(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\rangle\leq\langle\alpha(b)\rangle.$

Since
$\langle(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\rangle\leq\langle\alpha(b)\rangle$,
there exist $w\in B$ such that

$\|w\alpha(b)w^{*}-(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\|<\delta.$

Since $\|w\alpha(b)w^{*}-(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\|<\delta$, we
have

$\|\beta_{n}(w\alpha(b)w^{*})-\beta_{n}((\alpha((a-\delta)_{+}+d)-2\delta)_{+})\|<\delta.$

By $(4)$ we have

$\|\beta_{n}(w)\beta_{n}\alpha(b)\beta_{n}(w^{*})-\beta_{n}((\alpha((a-\delta)_{+}+d)-2\delta)_{+})\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle(\beta_{n}(\alpha((a-\delta)_{+}+d))-6\delta)_{+}\rangle\leq\langle\beta_{n}\alpha(b)\rangle.$

Since $(a-\delta)_{+}$ orthogonal to $d$, by $(3)$ and $(4)$, we may assume
that $\beta_{n}\alpha((a-\delta)_{+})$ orthogonal to $\beta_{n}\alpha(d)$.

With $G=\\{\gamma_{n}(a),\gamma_{n}(p_{0}),~{}\gamma_{n}(v_{i,j}):~{}1\leq
i\leq k,~{}1\leq j\leq k\\},$ and any $\varepsilon^{\prime\prime}>0$, let
$E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is asymptotically tracially in
$\Omega$, there exist a ${\rm C^{*}}$-algebra $D$ in $\Omega$ and completely
positive contractive linear maps $\alpha^{\prime}:E\to D$ and
$\beta_{n}^{\prime}:D\to E$, and $\gamma_{n}^{\prime}:E\to
E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1)^{\prime}$ the map $\alpha^{\prime}$ is unital completely positive linear
map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{E})$ are all
projections, $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{E})=1_{E}$ for
all $n\in\mathbb{N}$,

$(2)^{\prime}$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3)^{\prime}$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4)^{\prime}$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5)^{\prime}$ $\gamma_{n}^{\prime}(1)\lesssim\beta_{n}\alpha(d)$ for all
$n\in\mathbb{N}$.

By $(2)$, we have

$\displaystyle\|(\gamma_{n}\otimes id_{M_{k}}(v))({\rm
diag}(\gamma_{n}(p_{0})\otimes 1_{l},~{}0\otimes 1_{k-l}))(\gamma_{n}\otimes
id_{M_{k}}({v}^{*}))$ $\displaystyle-\gamma_{n}((a-\delta)_{+}+d)\otimes
1_{k}\|<\delta.$

Since $\|(\gamma_{n}\otimes id_{M_{k}}(v))({\rm diag}(\gamma_{n}(p_{0})\otimes
1_{l},~{}0\otimes 1_{k-l}))(\gamma_{n}\otimes
id_{M_{k}}({v}^{*}))-\gamma_{n}((a-\delta)_{+}+d)\otimes 1_{k}\|<\delta$, by
$(1)^{\prime}$ and $(3)^{\prime}$, we have

$\displaystyle\|\alpha^{\prime}\otimes id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v){\rm diag}(\alpha^{\prime}\gamma_{n}(p_{0})\otimes
1_{l},~{}0\otimes 1_{k-l}))$ $\displaystyle\alpha^{\prime}\otimes
id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v^{*})-\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)\otimes
1_{k}\|<2\delta.$

By Theorem 2.3 (1), we have

$k\langle(\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)-4\delta)_{+}\rangle\leq
l\langle(\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}\rangle.$

Since $B\in\Omega,$ this implies

$\langle(\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)-4\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}\rangle.$

Since
$\langle(\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)-4\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}\rangle,$
there exist $w\in D$ such that

$\|w(\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}w^{*}-\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)-4\delta)_{+}\|<\delta.$

By $(4)^{\prime}$, we have

$\|\beta_{n}^{\prime}(w)\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}\beta_{n}(w^{*})-\beta_{n}\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)-\delta)_{+}\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle(\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)-\delta)_{+}-2\delta)_{+}\rangle\leq\langle\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+})\rangle.$

Therefore, we have

$\displaystyle\langle(a-\varepsilon)_{+}\rangle$
$\displaystyle\leq\langle(\gamma_{n}(a)-8\delta)_{+}\rangle+\langle(\beta_{n}\alpha((a-\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(a)\rangle+\langle\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-\delta)_{+}-6\delta)_{+}))\rangle$
$\displaystyle+\langle(\beta_{n}\alpha((a-\delta)_{+})-2\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(1_{A})\rangle+\langle(\beta_{n}\alpha((a-\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle\leq\langle\beta_{n}\alpha(d)\rangle+\langle(\beta_{n}\alpha((a-\delta)_{+})-2\delta)_{+}\rangle+\langle\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-\delta)_{+})-4\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}\rangle+\langle\beta_{n}\alpha(p_{0})\rangle\leq\langle
p_{0}\rangle.$

$(\textbf{III})$, we suppose that both $a$ and $b$ are Cuntz equivalent to
projections.

Choose projections $p,~{}q$ such that $a$ is Cuntz equivalent to $p$ and $b$
is Cuntz equivalent to $q.$ We may assume that $a=p,~{}b=q.$ Since $k\langle
p\rangle\leq l\langle q\rangle$. Hence, by Theorem 2.3 (1), there exists
$v=(v_{i,j})\in{\rm M}_{k}(A),~{}1\leq i\leq k,~{}1\leq j\leq k$ such that

$v{\rm diag}(q\otimes 1_{l},~{}0\otimes 1_{k-l}){v}^{*}=p\otimes 1_{k}.$

Since $A\in{\rm ATA}\mathcal{P}$, for
$F=\\{p,~{}q,\gamma_{n}(v_{i,j}):~{}1\leq i\leq k,~{}1\leq j\leq k\\},$ for
any $\varepsilon^{\prime}>0,$ there exist a ${\rm C^{*}}$-algebra $B$ in
$\mathcal{P}$ and completely positive contractive linear maps $\alpha:A\to B$
and $\beta_{n}:B\to A$, and $\gamma_{n}:A\to A\cap\beta_{n}(B)^{\perp}$ such
that

$(1)$ the map $\alpha$ is unital completely positive linear map,
$\beta_{n}(1_{B})$ and $\gamma_{n}(1_{A})$ are all projections
$\beta_{n}(1_{B})+\gamma_{n}(1_{A})=1_{A}$ for all $n\in\mathbb{N}$,

$(2)$ $\|x-\gamma_{n}(x)-\beta_{n}(\alpha(x))\|<\varepsilon^{\prime}$ for all
$x\in F$ and for all $n\in{\mathbb{N}}$.

$(3)$ $\alpha$ is an $F$-$\varepsilon^{\prime}$ approximate embedding,

$(4)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(xy)\|=\|x\|$ for all $x,y\in B$.

Since $v{\rm diag}(q\otimes 1_{l},~{}0\otimes 1_{k-l})v^{*}=p\otimes 1_{k}$,
by $(2)$, we have

$\displaystyle\|(\gamma_{n}\otimes id_{M_{k}}(v)+\beta_{n}\otimes
id_{M_{k}}(\alpha\otimes id_{M_{k}}(v)))({\rm diag}(\gamma_{n}(q)\otimes
1_{l},~{}0\otimes 1_{k-l})$ $\displaystyle+{\rm
diag}(\beta_{n}\alpha(q)\otimes 1_{l},~{}0\otimes 1_{k-l}))(\gamma_{n}\otimes
id_{M_{k}}({v}^{*})+\beta_{n}\otimes id_{M_{k}}(\alpha\otimes
id_{M_{k}}({v}^{*})))$ $\displaystyle-(\gamma_{n}(p)\otimes
1_{k},+\beta_{n}\alpha(p)\otimes 1_{k})\|<\delta.$

By $(1)$, we have

$\displaystyle\|(\gamma_{n}\otimes id_{M_{k}}(v)){\rm
diag}(\gamma_{n}(q)\otimes 1_{l},~{}0\otimes 1_{k-l})(\gamma_{n}\otimes
id_{M_{k}}({v}^{*}))$ $\displaystyle-\gamma_{n}(p)\otimes 1_{k}\|<\delta.$

Since $v{\rm diag}(q\otimes 1_{l},~{}0\otimes 1_{k-l})v^{*}=p\otimes 1_{k}$,
by $(1)$ and $(3)$, we have

$\|\alpha\otimes id_{M_{k}}(v)){\rm diag}(\alpha(q)\otimes 1_{l},~{}0\otimes
1_{k-l})\alpha\otimes id_{M_{k}}({v}^{*})-\alpha(p)\otimes 1_{k}\|<\delta.$

Since $\|\alpha\otimes id_{M_{k}}(v)){\rm diag}(\alpha(q)\otimes
1_{l},~{}0\otimes 1_{k-l})\alpha\otimes id_{M_{k}}({v}^{*})-\alpha(p)\otimes
1_{k}\|<\delta,$ by Theorem 2.3 (1), we have

$k\langle(\alpha(p)-\delta)_{+}\rangle\leq l\langle\alpha(q)\rangle.$

Since $B\in\Omega$, we have

$\langle(\alpha(p)-\delta)_{+}\rangle\leq\langle\alpha(q)\rangle.$

Since $\langle(\alpha(p)-\delta)_{+}\rangle\leq\langle\alpha(q)\rangle,$ there
exists $w\in B$ such that

$\|w\alpha(q)w^{*}-(\alpha(p)-\delta)_{+}\|<\delta.$

By $(4)$, we have

$\|\beta_{n}(w)\beta_{n}\alpha(q)\beta_{n}(w^{*})-\beta_{n}((\alpha(p)-\delta)_{+})\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle\leq\langle\beta_{n}\alpha(q)\rangle.$

$(\textbf{III.I})$, if $(\beta_{n}\alpha(p)-2\delta)_{+}$ and
$\beta_{n}\alpha(q)$ are not Cuntz equivalent to a pure positive element, then
there exist projections $p_{1},q_{1}$ such that
$(\beta_{n}\alpha(p)-2\delta)_{+}\sim p_{1}$ and $\beta_{n}\alpha(q)\sim
q_{1}$, then we have $k\langle p_{1}\rangle\leq l\langle q_{1}\rangle=l\langle
p_{1}\rangle\leq k\langle p_{1}\rangle.$ So $\bigoplus_{n=1}^{k}p_{1}$ is
equivalent to a proper subprojection of itself, and this contradicts the
stable finiteness of $A$ (since ${\rm C^{*}}$-algebras in $\mathcal{P}$ are
stably finite (cf. proposition 4.2 in [10]). So we may assume that there exist
a nonzero projection $s\in A$ and orthogonal to
$(\beta_{n}\alpha(p)-2\delta)_{+}$ such that
$(\beta_{n}\alpha(p)-2\delta)_{+}+s\lesssim\beta_{n}\alpha(q)$.

With $G=\\{\gamma_{n}(p),\gamma_{n}(q),~{}\gamma_{n}(v_{i,j}):~{}1\leq i\leq
k,~{}1\leq j\leq k\\},$ and any $\varepsilon^{\prime\prime}>0$,
$E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is asymptotically tracially in
$\Omega$, there exist a ${\rm C^{*}}$-algebra $D$ in $\Omega$ and completely
positive contractive linear maps $\alpha^{\prime}:E\to D$ and
$\beta_{n}^{\prime}:D\to E$, and $\gamma_{n}^{\prime}:E\to
E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1)^{\prime}$ the map $\alpha^{\prime}$ is unital completely positive linear
map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{E})$ are all
projections, $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{E})=1_{E}$ for
all $n\in\mathbb{N}$,

$(2)^{\prime}$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3)^{\prime}$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4)^{\prime}$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5)^{\prime}$ $\gamma_{n}^{\prime}\gamma_{n}(1)\lesssim s$ for all
$n\in\mathbb{N}$.

Since

$\|(\gamma_{n}\otimes id_{M_{k}}(v)){\rm diag}(\gamma_{n}(q)\otimes
1_{l},~{}0\otimes 1_{k-l})(\gamma_{n}\otimes
id_{M_{k}}({v}^{*}))-\gamma_{n}(p)\otimes 1_{k}\|<\delta.$

By $(1)^{\prime}$, we have

$\displaystyle\|\alpha^{\prime}\otimes id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v)){\rm diag}(\alpha^{\prime}\gamma_{n}(q)\otimes 1_{l},~{}0\otimes
1_{k-l})$ $\displaystyle\alpha^{\prime}\otimes id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v^{*})-\alpha^{\prime}\gamma_{n}(p)\otimes 1_{k}\|<2\delta.$

By Theorem 2.3 (1), we have

$k\langle(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle\leq
l\langle(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

Since $D\in\Omega$, we have

$\langle(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

There exists $w\in D$ such that

$\|w(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}w^{*}-(\alpha^{\prime}\gamma_{n}(p)-2\delta)_{+}\|<\delta.$

By $(4^{\prime})$, we have

$\|\beta_{n}^{\prime}(w)\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\beta_{n}(w^{*})-\beta_{n}((\alpha^{\prime}\gamma_{n}(p)-2\delta)_{+})\|<2\delta.$

We have

$\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-6\delta)_{+}\rangle\leq\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

Therefore, if $\varepsilon^{\prime}$, are small enough, then

$\displaystyle\langle(p-\varepsilon)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}(p)-6\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle$
$\displaystyle\leq\langle(\gamma_{n}^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)\rangle$
$\displaystyle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(1_{A})\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle$
$\displaystyle\leq\langle
s\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-2\delta)_{+}\rangle\leq\langle
q\rangle.$

$(\textbf{III.II})$, We suppose that $(\beta_{n}\alpha(p)-2\delta)_{+}$ is a
purely positive element, then, by Theorem 2.3 (2), there is a non-zero
positive element $d$ orthogonal to $(\beta_{n}\alpha(p)-2\delta)_{+}$ such
that
$((\beta_{n}\alpha(p)-2\delta)_{+}-\delta)_{+}+d\lesssim(\beta_{n}\alpha(p)-2\delta)_{+}$.

With $G=\\{\gamma_{n}(p),\gamma_{n}(q),~{}\gamma_{n}(v_{i,j}):~{}1\leq i\leq
k,~{}1\leq j\leq k\\},$ and $\varepsilon^{\prime\prime}>0$,
$E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is asymptotically tracially in
$\Omega$, there exist a ${\rm C^{*}}$-algebra $D$ in $\Omega$ and completely
positive contractive linear maps $\alpha^{\prime}:E\to D$ and
$\beta_{n}^{\prime}:D\to E$, and $\gamma_{n}^{\prime}:E\to
E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1)^{\prime}$ the map $\alpha^{\prime}$ is unital completely positive linear
map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{E})$ are all
projections, $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{E})=1_{E}$ for
all $n\in\mathbb{N}$,

$(2)^{\prime}$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3)^{\prime}$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4)^{\prime}$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5)^{\prime}$ $\gamma_{n}^{\prime}\gamma_{n}(1)\lesssim d$ for all
$n\in\mathbb{N}$.

Since

$\|(\gamma_{n}\otimes id_{M_{k}}(v)){\rm diag}(\gamma_{n}(q)\otimes
1_{l},~{}0\otimes 1_{k-l})(\gamma_{n}\otimes
id_{M_{k}}({v}^{*}))-\gamma_{n}(p)\otimes 1_{k}\|<\delta.$

By $(1)^{\prime}$, we have

$\displaystyle\|\alpha^{\prime}\otimes id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v){\rm diag}(\alpha^{\prime}\gamma_{n}(q)\otimes 1_{l},~{}0\otimes
1_{k-l})$ $\displaystyle\alpha^{\prime}\otimes id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v^{*})-\alpha^{\prime}\gamma_{n}(p)\otimes 1_{k}\|<2\delta.$

By Theorem 2.3 (1), we have

$k\langle(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle\leq
l\langle(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

Since $D\in\Omega$, we have

$\langle(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

There exists $w\in D$ such that

$\|w(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}w^{*}-(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\|<\delta.$

By $(4^{\prime})$, we have

$\|\beta_{n}^{\prime}(w)\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\beta_{n}(w^{*})-\beta_{n}((\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+})\|<2\delta.$

We have

$\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-8\delta)_{+}\rangle\leq\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

Therefore, if $\varepsilon^{\prime}$, are small enough, then

$\displaystyle\langle(p-\varepsilon)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}(p)-6\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle$
$\displaystyle\leq\langle(\gamma_{n}^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)\rangle$
$\displaystyle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(1_{A})\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle$
$\displaystyle\leq\langle
s\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-2\delta)_{+}\rangle$
$\displaystyle\leq\langle q\rangle.$

$(\textbf{III.III})$, we suppose that $(\beta_{n}\alpha(p)-2\delta)_{+}$ is
Cuntz equivalent to a projection and $\beta_{n}\alpha(q)$ is not Cuntz
equivalent to a projection. Choose a projection $p_{0}$ such that
$(\beta_{n}\alpha(p)-2\delta)_{+}$ is Cuntz equivalent to $p_{0}.$ We may
assume that $(\beta_{n}\alpha(p)-2\delta)_{+}=p_{0}.$

By Theorem 2.3 (2), there exists a non-zero positive element $d$ orthogonal to
$\beta_{n}\alpha(q)$ such that
$\langle(\beta_{n}\alpha(q)-4\delta)_{+}+d\rangle\leq\langle\beta_{n}\alpha(q)\rangle.$

With $G=\\{\gamma_{n}(p),\gamma_{n}(q),~{}\gamma_{n}(v_{i,j}):~{}1\leq i\leq
k,~{}1\leq j\leq k\\},$ and any sufficiently small
$\varepsilon^{\prime\prime}>0$, $E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is
asymptotically tracially in $\Omega$, there exist a ${\rm C^{*}}$-algebra $D$
in $\Omega$ and completely positive contractive linear maps
$\alpha^{\prime}:E\to D$ and $\beta_{n}^{\prime}:D\to E$, and
$\gamma_{n}^{\prime}:E\to E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1)^{\prime}$ the map $\alpha^{\prime}$ is unital completely positive linear
map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{E})$ are all
projections, $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{E})=1_{E}$ for
all $n\in\mathbb{N}$,

$(2)^{\prime}$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3)^{\prime}$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4)^{\prime}$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5)^{\prime}$ $\gamma_{n}^{\prime}\gamma_{n}(1)\lesssim d$ for all
$n\in\mathbb{N}$.

Since

$\|(\gamma_{n}\otimes id_{M_{k}}(v)){\rm diag}(\gamma_{n}(q)\otimes
1_{l},~{}0\otimes 1_{k-l})(\gamma_{n}\otimes
id_{M_{k}}({v}^{*}))-\gamma_{n}(p)\otimes 1_{k}\|<\delta.$

By $(1)^{\prime}$, we have

$\displaystyle\|\alpha^{\prime}\otimes id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v){\rm diag}(\alpha^{\prime}\gamma_{n}(q)\otimes 1_{l},~{}0\otimes
1_{k-l})$ $\displaystyle\alpha^{\prime}\otimes id_{M_{k}}\gamma_{n}\otimes
id_{M_{k}}(v^{*})-\alpha^{\prime}\gamma_{n}(p)\otimes 1_{k}\|<2\delta.$

By Theorem 2.3 (1), we have

$k\langle(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle\leq
l\langle(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

Since $D\in\Omega$, we have

$\langle(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

There exists $w\in D$ such that

$\|w(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}w^{*}-(\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\|<\delta.$

By $(4^{\prime})$, we have

$\|\beta_{n}^{\prime}(w)\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\beta_{n}(w^{*})-\beta_{n}((\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+})\|<2\delta.$

We have

$\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-8\delta)_{+}\rangle\leq\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q)-\delta)_{+}\rangle.$

Therefore, we have

$\displaystyle\langle(p-\varepsilon)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}(p)-6\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle$
$\displaystyle\leq\langle(\gamma_{n}^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)\rangle$
$\displaystyle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(1_{A})\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-4\delta)_{+}\rangle$
$\displaystyle\leq\langle
s\rangle+\langle(\beta_{n}\alpha(p)-2\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-2\delta)_{+}\rangle$
$\displaystyle\leq\langle q\rangle.$

∎

######  Theorem 3.2.

Let $\mathcal{P}$ be a class of stably finite unital ${\rm C^{*}}$-algebras
which have Winter’s $n$-comparison. Then $A$ has Winter’s $n$-comparison for
any simple unital ${\rm C^{*}}$-algebra $A\in{\rm ATA}\mathcal{P}.$

###### Proof.

As in the proof of Theorem 3.1, we may suppose that $\mathcal{P}$ is closed
under Morita equivalence.

Let $a,b_{0},b_{1},\cdots,b_{n}\in{\rm M}_{\infty}(A)_{+}$. By Theorem 2.3
(3), we need only to show that $\langle(a-\varepsilon)_{+}\rangle\leq\langle
b_{0}\rangle+\langle b_{1}\rangle+\cdots+\langle b_{n}\rangle$ for any
$\varepsilon>0$, if $(k_{i}+1)\langle a\rangle\leq k_{i}\langle
b_{i}\rangle,~{}0\leq i\leq n.$ Note that $k_{i}$ can be chosen to be the same
for all $b_{i},$ as, with $k=(k_{0}+1)(k_{1}+1)\cdots(k_{n}+1)-1,$ one has
$(k+1)\langle a\rangle\leq k\langle b_{i}\rangle$ for all $0\leq i\leq n.$

We may assume that $a,b_{0},b_{1},\cdots,b_{n}\in{\rm M}_{k}(A)_{+}$ for some
sufficiently large integer $k$. By Lemma 2.4, and Lemma 2.5, we may assume
that $a,b_{0},b_{1},\cdots,b_{n}\in A_{+}$.

We divide the proof into three cases.

$(\textbf{I})$, let us suppose that $b_{i}$ is not Cuntz equivalent to a
projection for some $0\leq i\leq n.$ We may assume that $b_{0}$ is a purely
positive element. By Theorem 2.3 (2), for $\delta>0$, there is a non-zero
positive element $d$ orthogonal to $b_{0}$ such that
$(b_{0}-\delta/2)_{+}+d\lesssim b_{0}$. Hence, as the proof of Theorem 3.1, by
Theorem 2.3 (1), there exist $v_{k}=(v_{i,j}^{k}),~{}0\leq k\leq n,~{}1\leq
i\leq k+1,~{}1\leq j\leq k+1$ such that

$\|v_{i}{\rm diag}(b_{i}\otimes 1_{k},~{}0)v_{i}^{*}-a\otimes
1_{k+1}\|<\delta,$

where $0\leq i\leq n.$

Since $A\in{\rm ATA}\mathcal{P}$, for
$F=\\{a,~{}b_{0},~{}b_{1},~{}\cdots,~{}b_{n},~{}d,~{}{v_{i,j}^{k}}:~{}1\leq
i\leq k+1,~{}1\leq j\leq k+1,~{}0\leq k\leq n\\},$ and
$\varepsilon^{\prime}>0$, there exist a ${\rm C^{*}}$-algebra $B$ in
$\mathcal{P}$ and completely positive contractive linear maps $\alpha:A\to B$
and $\beta_{n}:B\to A$, and $\gamma_{n}:A\to A\cap\beta_{n}(B)^{\perp}$ such
that

$(1)$ the map $\alpha$ is unital completely positive linear map,
$\beta_{n}(1_{B})$ and $\gamma_{n}(1_{A})$ are all projections
$\beta_{n}(1_{B})+\gamma_{n}(1_{A})=1_{A}$ for all $n\in\mathbb{N}$,

$(2)$ $\|x-\gamma_{n}(x)-\beta_{n}(\alpha(x))\|<\varepsilon^{\prime}$ for all
$x\in F$ and for all $n\in{\mathbb{N}}$,

$(3)$ $\alpha$ is an $F$-$\varepsilon^{\prime}$ approximate embedding,

$(4)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(xy)\|=\|x\|$ for all $x,y\in B$, and

$(5)$ $\gamma_{n}(1)\lesssim d$ for all $n\in\mathbb{N}$.

Since

$\|v_{i}{\rm diag}(b_{i}\otimes 1_{k},~{}0)v_{i}^{*}-a\otimes
1_{k+1}\|<\delta,$

for all $0\leq i\leq n.$

By $(3)$, we have

$\|\alpha\otimes id_{M_{k+1}}(v_{i}){\rm diag}(\alpha(b_{i})\otimes
1_{k},~{}0))\alpha\otimes id_{M_{k+1}}({v_{i}}^{*})-\alpha(a)\otimes
1_{k+1}\|<\delta,$

for all $0\leq i\leq n.$

By Theorem 2.3 (1), we have

$(k+1)\langle(\alpha(a)-\delta)_{+}\rangle\leq\langle\alpha(b_{i})\rangle,$

for all $0\leq i\leq n.$

Since $B\in\Omega,$ we have

$\langle(\alpha(a)-\delta)_{+}\rangle\leq\langle\alpha(b_{0})\rangle+\langle\alpha(b_{1})\rangle+\cdots+\langle\alpha(b_{n})\rangle.$

Since

$\langle(\alpha(a)-\delta)_{+}\rangle\leq\langle\alpha(b_{0})\rangle+\langle\alpha(b_{1})\rangle+\cdots+\langle\alpha(b_{n})\rangle,$

There exists $w\in M_{k+1}(D)$ such that

$\|w{\rm
diag}(\alpha(b_{0}),\alpha(b_{1}),\cdots,\alpha(b_{n}))w^{*}-(\alpha(a)-\delta)_{+}\|<\delta.$

By $(4)$, we have

$\|\beta_{n}\otimes id_{M_{k+1}}(w){\rm
diag}(\beta_{n}\alpha(b_{0}),\beta_{n}\alpha(b_{1}),\cdots,\beta_{n}\alpha(b_{n}))\beta_{n}\otimes
id_{M_{k+1}}(w^{*})$ $-(\beta_{n}\alpha(a)-\delta)_{+}\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle(\beta_{n}\alpha(a)-4\delta)_{+}\rangle\leq\langle\beta_{n}\alpha(b_{0}-\delta)_{+}\rangle+\langle\beta_{n}\alpha(b_{1})\rangle+\cdots+\langle\beta_{n}\alpha(b_{n})\rangle,$

Therefore, we have

$\displaystyle\langle(a-\varepsilon)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}(a)-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(a)-4\delta)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}(a)\rangle+\langle\beta_{n}\alpha(b_{0}-\delta)_{+})\rangle+\langle\beta_{n}\alpha(b_{1})\rangle$
$\displaystyle+\cdots+\langle\beta_{n}\alpha(b_{n})\rangle\leq\langle\gamma_{n}(1_{A})\rangle+\langle\beta_{n}\alpha(b_{0}-\delta)_{+}\rangle$
$\displaystyle+\langle\beta_{n}\alpha(b_{1})\rangle+\cdots+\langle\beta_{n}\alpha(b_{n})\rangle$
$\displaystyle\leq\langle
d\rangle+\langle\beta_{n}\alpha(b_{0}-\delta)_{+}\rangle$
$\displaystyle+\langle\beta_{n}\alpha(b_{1})\rangle+\cdots+\langle\beta_{n}\alpha(b_{n})\rangle$
$\displaystyle\leq\langle(b_{0}-\delta/2)_{+}\rangle+\langle d\rangle+\langle
b_{1}\rangle+\cdots+\langle b_{n}\rangle\leq\langle b_{0}\rangle+\langle
b_{1}\rangle+\cdots+\langle b_{n}\rangle.$

$(\textbf{II})$, let us suppose that each $b_{i}~{}(0\leq i\leq n)$ is Cuntz
equivalent to a projection and $a$ is not Cuntz equivalent to a projection.
Choose projections $p_{0},~{}p_{1},~{}\cdots,~{}p_{n}$ such that $b_{i}$ is
Cuntz equivalent to $p_{i}$ for all $0\leq i\leq n.$ We may assume that
$b_{i}=p_{i}$ for all $0\leq i\leq n.$

By Theorem 2.3 (2), let $\delta>0$ (with $\delta<\varepsilon$), there is a
non-zero positive element $d$ orthogonal to $a$ such that
$\langle(a-\delta)_{+}+d\rangle\leq\langle a\rangle$. Since $(k+1)\langle
a\rangle\leq k\langle p_{i}\rangle,$ we have
$(k+1)\langle(a-\delta)_{+}+d\rangle\leq k\langle p_{i}\rangle$ for all $0\leq
i\leq n.$

Hence, by Theorem 2.3 (1), as earlier, there exist $v_{k}=(v_{i,j}^{k})\in{\rm
M}_{k+1}(A),$ $0\leq k\leq n,~{}1\leq i\leq k+1,~{}1\leq j\leq k+1$, such that

$\|v_{i}{\rm diag}(p_{i}\otimes 1_{k},~{}0)v_{i}^{*}-((a-\delta)_{+}+d)\otimes
1_{k+1}\|<\delta,~{}0\leq i\leq n.$

Since $A\in{\rm ATA}\mathcal{P},$ for
$F=\\{a,~{}b_{0},~{}b_{1},~{}\cdots,~{}b_{n},~{}d,~{}{v_{i,j}^{k}}:~{}1\leq
i\leq k+1,~{}1\leq j\leq k+1,~{}0\leq k\leq n\\},$ for any
$\varepsilon^{\prime}>0$, there exist a ${\rm C^{*}}$-algebra $B$ in
$\mathcal{P}$ and completely positive contractive linear maps $\alpha:A\to B$
and $\beta_{n}:B\to A$, and $\gamma_{n}:A\to A\cap\beta_{n}(B)^{\perp}$ such
that

$(1)$ the map $\alpha$ is unital completely positive linear map,
$\beta_{n}(1_{B})$ and $\gamma_{n}(1_{A})$ are all projections
$\beta_{n}(1_{B})+\gamma_{n}(1_{A})=1_{A}$ for all $n\in\mathbb{N}$,

$(2)$ $\|x-\gamma_{n}(x)-\beta_{n}(\alpha(x))\|<\varepsilon^{\prime}$ for all
$x\in F$ and for all $n\in{\mathbb{N}}$.

$(3)$ $\alpha$ is an $F$-$\varepsilon^{\prime}$ approximate embedding,

$(4)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(xy)\|=\|x\|$ for all $x,y\in B$.

Since $\|v_{i}{\rm diag}(p_{i}\otimes
1_{k},~{}0)v_{i}^{*}-((a-\delta)_{+}+d)\otimes 1_{k+1}\|<\delta,~{}0\leq i\leq
n$, By $(2)$, $(3)$ and $(4)$, we have

$\displaystyle\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i})+\beta_{n}\otimes
id_{M_{k+1}}(\alpha\otimes id_{M_{k}}(v_{i})))({\rm
diag}(\gamma_{n}(p_{i})\otimes 1_{k},~{}0)$ $\displaystyle+{\rm
diag}(\beta_{n}\alpha(p_{i})\otimes 1_{k},~{}0))(\gamma_{n}\otimes
id_{M_{k+1}}({v_{i}}^{*})+\beta_{n}\otimes id_{M_{k+1}}(\alpha\otimes
id_{M_{k+1}}({v_{i}}^{*})))$
$\displaystyle-(\gamma_{n}((a-\delta)_{+}+d)\otimes
1_{k+},+\beta_{n}\alpha((a-\delta)_{+}+d)\otimes 1_{k+1})\|<\delta.$

By $(1)$, we have

$\displaystyle\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i}))({\rm
diag}(\gamma_{n}(p_{i})\otimes 1_{k},~{}0))(\gamma_{n}\otimes
id_{M_{k+1}}({v_{i}}^{*}))$ $\displaystyle-\gamma_{n}((a-\delta)_{+}+d)\otimes
1_{k+1}\|<\delta.$

Since $\|v_{i}{\rm diag}(p_{i}\otimes
1_{k},~{}0)v_{i}^{*}-((a-\delta)_{+}+d)\otimes 1_{k+1}\|<\delta,~{}0\leq i\leq
n$, by $(1)$ and $(3)$, we have

$\|\alpha\otimes id_{M_{k+1}}(v)){\rm diag}(\alpha(p_{i})\otimes
1_{l},~{}0)\alpha\otimes id_{M_{k+1}}({v}^{*})-\alpha((a-\delta)_{+}+d)\otimes
1_{k+1}\|<2\delta,$

for $0\leq i\leq n$.

By Theorem 2.3 (1), we have

$(k+1)\langle(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\rangle\leq
k\langle\alpha(p_{i})\rangle$

for $0\leq i\leq n$.

Since $B\in\Omega$, we have

$\langle(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\rangle\leq\langle\alpha(p_{0})\rangle+\langle\alpha(p_{1})\rangle+\cdots+\langle\alpha(p_{n})\rangle.$

Since
$\langle(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\rangle\leq\langle\alpha(p_{0})\rangle+\langle\alpha(p_{1})\rangle+\cdots+\langle\alpha(p_{n})\rangle,$
there exists $w\in M_{k+1}(B)$ such that

$\|w{\rm
diag}(\alpha(p_{0}),\alpha(p_{1}),\cdots,\alpha(p_{n}))w^{*}-(\alpha((a-\delta)_{+}+d)-2\delta)_{+}\|<\delta.$

By $(4)$, we have

$\|\beta_{n}\otimes id_{M_{k+1}}(w){\rm
diag}(\beta_{n}\alpha(p_{0}),\beta_{n}\alpha(p_{1}),\cdots,\beta_{n}\alpha(p_{n}))\beta_{n}\otimes
id_{M_{k+1}}(w^{*})$
$-(\beta_{n}(\alpha((a-\delta)_{+}+d)-2\delta)_{+})\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle(\beta_{n}(\alpha((a-\delta)_{+}+d)-6\delta)_{+})\rangle\leq\langle(\beta_{n}\alpha(p_{0})-\delta)_{+}\rangle+\langle\beta_{n}\alpha(p_{1})\rangle+\cdots+\langle\beta_{n}\alpha(p_{n})\rangle.$

Since $(a-\delta)_{+}$ is orthogonal to $d$, by $(3)$ and $(4)$, we may assume
that $\beta_{n}\alpha((a-\delta)_{+})$ is orthogonal to $\beta_{n}\alpha(d)$.

With $G=\\{\gamma_{n}(a),\gamma_{n}(p_{i}),~{}\gamma_{n}(v_{i,j}^{k}):~{}1\leq
i\leq k+1,~{}1\leq j\leq k+1,0\leq k\leq n\\},$ and any
$\varepsilon^{\prime\prime}>0$, $E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is
asymptotically tracially in $\mathcal{P}$, there exist a ${\rm C^{*}}$-algebra
$D$ in $\mathcal{P}$ and completely positive contractive linear maps
$\alpha^{\prime}:E\to D$ and $\beta_{n}^{\prime}:D\to E$, and
$\gamma_{n}^{\prime}:E\to E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1^{\prime})$ the map $\alpha^{\prime}$ is unital completely positive linear
map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{A})$ are all
projections, and $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{A})=1_{A}$
for all $n\in\mathbb{N}$,

$(2^{\prime})$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3^{\prime})$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4^{\prime})$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5^{\prime})$ $\gamma_{n}^{\prime}\gamma_{n}(1)\lesssim\beta_{n}\alpha(d)$
for all $n\in\mathbb{N}$.

Since

$\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i})){\rm diag}(\gamma_{n}(p_{i})\otimes
1_{k},~{}0)(\gamma_{n}\otimes id_{M_{k}}({v}^{*}))-\gamma_{n}(a)\otimes
1_{k+1}\|<\delta,$

for $0\leq i\leq n$, by $(1)^{\prime}$, we have

$\|\alpha^{\prime}\otimes id_{M_{k+1}}\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}){\rm diag}(\alpha^{\prime}\gamma_{n}(p_{i})\otimes
1_{k},~{}0)$ $\alpha^{\prime}\otimes id_{M_{k+1}}(\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}^{*}))-\alpha^{\prime}\gamma_{n}(a)\otimes
1_{k+1}\|<2\delta.$

By Theorem 2.3 (1), we have

$(k+1)\langle(\alpha^{\prime}\gamma_{n}(a)-4\delta)_{+}\rangle\leq
k\langle\alpha^{\prime}\gamma_{n}(p_{i})\rangle.$

Since $D\in\Omega$, we have

$\langle(\alpha^{\prime}\gamma_{n}(a)-4\delta)_{+}\rangle\leq\langle\alpha^{\prime}\gamma_{n}(p_{0})\rangle+\alpha^{\prime}\gamma_{n}(p_{1})\rangle+\cdots+\alpha^{\prime}\gamma_{n}(p_{n})\rangle.$

Since
$\langle(\alpha^{\prime}\gamma_{n}(a)-4\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}\rangle+(\alpha^{\prime}\gamma_{n}(p_{1})-\delta)_{+}\rangle+\cdots+(\alpha^{\prime}\gamma_{n}(p_{n})-\delta)_{+}\rangle,$
there exists $w\in M_{k+1}(D)$ such that

$\|w{\rm
diag}(\alpha^{\prime}\gamma_{n}(p_{0}),\alpha^{\prime}\gamma_{n}(p_{1}),\cdots,\alpha^{\prime}\gamma_{n}(p_{n}))w^{*}-(\alpha^{\prime}\gamma_{n}((a-\delta)_{+}+d)-2\delta)_{+}\|<\delta.$

By $(4)$, we have

$\|\beta_{n}^{\prime}\otimes id_{M_{k+1}}(w){\rm
diag}(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{0}),\beta_{n}^{\prime}\alpha^{\prime}\beta_{n}\alpha(p_{1}),\cdots,\beta_{n}^{\prime}\alpha^{\prime}\beta_{n}\alpha(p_{n}))\beta_{n}\otimes
id_{M_{k+1}}(w^{*})$
$-(\beta_{n}^{\prime}\alpha^{\prime}\beta_{n}(\alpha((a-\delta-2\delta)_{+})\|<2\delta.$

By Theorem 2.3 (1), we have

$\|\langle(\beta_{n}^{\prime}(\alpha^{\prime}((a-\delta)_{+}-2\delta)_{+})\rangle\leq\langle\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{0})\rangle+\langle\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{1})\rangle+\cdots+\langle\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{n})\rangle.$

Therefore, we have

$\displaystyle\langle(a-\varepsilon)_{+}\rangle$
$\displaystyle\leq\langle(\gamma_{n}(a)-10\delta)_{+}\rangle+\langle(\beta_{n}\alpha(a-\delta)_{+})-8\delta)_{+}\rangle$
$\displaystyle\leq\langle(\gamma_{n}^{\prime}\gamma_{n}(a)-2\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-\delta)_{+})-8\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}\alpha((a-\delta)_{+})-8\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}(1_{A})\rangle$
$\displaystyle+\langle(\beta_{n}\alpha((a-\delta)_{+})-8\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle\leq\langle\beta_{n}\alpha(d)\rangle+\langle(\beta_{n}\alpha((a-\delta)_{+})-2\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(a)-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p_{1})\rangle+\cdots+\langle(\beta_{n}\alpha(p_{n})\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{0})-\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(p_{0})-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}\alpha(p_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle p_{0}\rangle+\langle p_{1}\rangle+\cdots+\langle
p_{n}\rangle.$

$(\textbf{III})$, we suppose that both $a$ and $b_{i}$ ($0\leq i\leq n$) are
all Cuntz equivalent to projections.

Choose projections $p,~{}q_{i}$ such that $a$ is Cuntz equivalent to $p$ and
$b_{i}$ is Cuntz equivalent to $q_{i}.$ We may assume that
$a=p,~{}b_{i}=q_{i}.$

Since $(k+1)\langle p\rangle\leq k\langle q_{i}\rangle$ for all $0\leq i\leq
n,$ so for any $\delta>0$, by Theorem 2.3 (1), there exist
$v_{k}=(v_{i,j}^{k}),~{}0\leq k\leq n,~{}1\leq i\leq k+1,~{}1\leq j\leq k+1$
such that

$\|v_{i}{\rm diag}(q_{i}\otimes 1_{k},~{}0)v_{i}^{*}-p\otimes
1_{k+1}\|<\delta.$

Since $A\in{\rm ATA}\mathcal{P}$, for $F=\\{p,~{}q_{k},(v_{i,j}^{k}),~{}0\leq
k\leq n,~{}1\leq i\leq k+1,~{}1\leq j\leq k+1\\},$ for any
$\varepsilon^{\prime}>0,$ there exist a a ${\rm C^{*}}$-algebra $B$ in
$\mathcal{P}$ and completely positive contractive linear maps $\alpha:A\to B$
and $\beta_{n}:B\to A$, and $\gamma_{n}:A\to A\cap\beta_{n}(B)^{\perp}$ such
that

$(1)$ the map $\alpha$ is unital completely positive linear map,
$\beta_{n}(1_{B})$ and $\gamma_{n}(1_{A})$ are all projections
$\beta_{n}(1_{B})+\gamma_{n}(1_{A})=1_{A}$ for all $n\in\mathbb{N}$,

$(2)$ $\|x-\gamma_{n}(x)-\beta_{n}(\alpha(x))\|<\varepsilon^{\prime}$ for all
$x\in F$ and for all $n\in{\mathbb{N}}$.

$(3)$ $\alpha$ is an $F$-$\varepsilon^{\prime}$ approximate embedding,

$(4)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(xy)\|=\|x\|$ for all $x,y\in B$.

Since $\|v_{i}{\rm diag}(q_{i}\otimes 1_{k},~{}0)v_{i}^{*}-p\otimes
1_{k+1}\|<\delta$ for $0\leq i\leq n$, by $(2)$, we have

$\displaystyle\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i})+\beta_{n}\otimes
id_{M_{k+1}}(\alpha\otimes id_{M_{k}}(v_{i})))({\rm
diag}(\gamma_{n}(q_{i})\otimes 1_{k},~{}0)$ $\displaystyle+{\rm
diag}(\beta_{n}\alpha(q_{i})\otimes 1_{k},~{}0))(\gamma_{n}\otimes
id_{M_{k+1}}({v_{i}}^{*})+\beta_{n}\otimes id_{M_{k+1}}(\alpha\otimes
id_{M_{k+1}}({v_{i}}^{*})))$ $\displaystyle-(\gamma_{n}(p)\otimes
1_{k+1}+\beta_{n}\alpha(p)\otimes 1_{k+1})\|$ $\displaystyle<2\delta.$

By $(1)$, we have

$\displaystyle\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i}))({\rm
diag}(\gamma_{n}(q_{i})\otimes 1_{k},~{}0))(\gamma_{n}\otimes
id_{M_{k+1}}({v_{i}}^{*}))$ $\displaystyle-\gamma_{n}(p)\otimes
1_{k+1}\|<2\delta.$

Since $\|v_{i}{\rm diag}(q_{i}\otimes 1_{k},~{}0)v_{i}^{*}-p\otimes
1_{k+1}\|<\delta$ for $0\leq i\leq n$, by $(1)$ and $(3)$, we have

$\|\alpha\otimes id_{M_{k+1}}(v_{i}){\rm diag}(\alpha(q_{i})\otimes
1_{k},~{}0)\alpha\otimes id_{M_{k+1}}({v_{i}}^{*})-\alpha(p)\otimes
1_{k+1}\|<\delta.$

By Theorem 2.3 (1), we have

$(k+1)\langle(\alpha(p)-\delta)_{+}\rangle\leq k\langle\alpha(q_{i})\rangle.$

Since $B\in\Omega$, we have

$\langle(\alpha(p)-\delta)_{+}\rangle\leq\langle\alpha(q_{0})\rangle+\langle\alpha(q_{1})\rangle+\cdots\
\langle\alpha(q_{n})\rangle.$

Since$\langle(\alpha(p)-\delta)_{+}\rangle\leq\langle\alpha(q_{0})\rangle+\langle\alpha(q_{1})\rangle+\cdots\
\langle\alpha(q_{n})\rangle,$ there exists $w\in M_{k+1}(D)$ such that

$\|w{\rm
diag}(\alpha(p_{0}),\alpha(p_{1}),\cdots,\alpha(p_{n}))w^{*}-(\alpha((a-\delta)_{+})\|<\delta.$

By $(4)$, we have

$\|\beta_{n}\otimes id_{M_{k+1}}(w){\rm
diag}(\beta_{n}\alpha(p_{0}),\beta_{n}\alpha(p_{1}),\cdots,\beta_{n}\alpha(p_{n}))\beta_{n}\otimes
id_{M_{k+1}}(w^{*})$
$-(\beta_{n}(\alpha((a-\delta)_{+})-2\delta)_{+})\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle(\beta_{n}(\alpha((a-\delta)_{+})-2\delta)_{+})\rangle\leq\langle\beta_{n}\alpha(p_{0})\rangle+\langle\beta_{n}\alpha(p_{1})\rangle+\cdots+\langle\beta_{n}\alpha(p_{n})\rangle.$

$(\textbf{III.I})$, if $(\beta_{n}\alpha(p)-2\delta)_{+}$ and
$\beta_{n}\alpha(q_{i})$ are not Cuntz equivalent to a pure positive element,
then there exist projections $p^{\prime},q^{\prime}_{i}$ such that
$(\beta_{n}\alpha(p)-2\delta)_{+}\sim p^{\prime}$ and
$\beta_{n}\alpha(q_{i})\sim q^{\prime}_{i}$, and we suppose that
$p^{\prime}\sim q_{0}^{\prime}+q_{1}^{\prime}+\cdots+q_{n}^{\prime}$, then we
have $(k+1)\langle p^{\prime}\rangle=(k+1)\langle
q_{0}^{\prime}\rangle+(k+1)\langle q_{1}^{\prime}\rangle+\cdots+(k+1)\langle
q_{n}^{\prime}\rangle\leq k\langle q_{0}^{\prime}\rangle+k\langle
q_{1}^{\prime}\rangle+\cdots+k\langle q_{n}^{\prime}\rangle,$ and this
contradicts the stable finiteness of $A$ (since ${\rm C^{*}}$-algebras in
$\mathcal{P}$ are stably finite (cf. proposition 4.2 in [10]). So there exist
a nonzero projection $s\in A$, orthogonal to $p^{\prime}$ such that

$(\beta_{n}\alpha(p)-2\delta)_{+}+s\lesssim
q_{0}^{\prime}+q_{1}^{\prime}+\cdots+q_{n}^{\prime}.$

With $G=\\{\gamma_{n}(a),\gamma_{n}(p_{i}),~{}\gamma_{n}(v_{i,j}^{k}):~{}1\leq
i\leq k+1,~{}1\leq j\leq k+1,0\leq k\leq n\\},$ and any
$\varepsilon^{\prime\prime}>0$, $E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is
asymptotically tracially in $\mathcal{P}$, there exist a ${\rm C^{*}}$-algebra
$D$ in $\Omega$ and completely positive contractive linear maps
$\alpha^{\prime}:E\to D$ and $\beta_{n}^{\prime}:D\to E$, and
$\gamma_{n}^{\prime}:E\to E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1)^{\prime}$ the map $\alpha^{\prime}$ is unital completely positive linear
map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{E})$ are all
projections, $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{E})=1_{E}$ for
all $n\in\mathbb{N}$,

$(2)^{\prime}$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3)^{\prime}$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4)^{\prime}$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5)^{\prime}$ $\gamma_{n}^{\prime}\gamma_{n}(1)\lesssim s$ for all
$n\in\mathbb{N}$.

Since

$\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i})){\rm diag}(\gamma_{n}(q_{i})\otimes
1_{k},~{}0)(\gamma_{n}\otimes id_{M_{k}}({v}^{*}))-\gamma_{n}(p)\otimes
1_{k+1}\|<2\delta,$

for $0\leq i\leq n$, by $(1)^{\prime}$, we have

$\|\alpha^{\prime}\otimes id_{M_{k+1}}\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}){\rm diag}(\alpha^{\prime}\gamma_{n}(q_{i})\otimes
1_{k},~{}0)$ $\alpha^{\prime}\otimes id_{M_{k+1}}(\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}^{*}))-\alpha^{\prime}\gamma_{n}(p)\otimes
1_{k+1}\|<3\delta.$

By Theorem 2.3 (1), we have

$(k+1)\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq
k\langle(\alpha^{\prime}\gamma_{n}(q_{i})-\delta)_{+}\rangle.$

Since $D\in\Omega$, we have

$\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle.$

Since
$\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle,$
there exists $w\in M_{k+1}(D)$ such that

$\|w{\rm
diag}((\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+},(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+},\cdots,(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+})w^{*}-(\alpha^{\prime}\gamma_{n}(p)-(n+3)\delta)_{+}\|<\delta.$

By $(4)$, we have

$\|\beta_{n}^{\prime}\otimes id_{M_{k+1}}(w){\rm
diag}((\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+},(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+},\cdots,(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+})$
$\beta_{n}^{\prime}\otimes
id_{M_{k+1}}(w^{*})-(\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(p)-(n+3)\delta)_{+}\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(p)-(n+10)\delta)_{+}\rangle\leq\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle.$

Therefore, we have

$\displaystyle\langle
p\rangle\leq\langle(\gamma_{n}(p)-(n+10)\delta)_{+})\rangle+\langle(\beta_{n}\alpha(p)-(n+10)\delta)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(p)\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-(n+12)\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p)-3\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(1_{A})\rangle$
$\displaystyle+\langle(\beta_{n}\alpha((a-\delta)_{+})-3\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-(n+8)\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle\leq\langle
s\rangle+\langle(\beta_{n}\alpha(p)-3\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-(n+10)\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}\alpha(q_{n})-\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}\alpha(q_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle q_{0}\rangle+\langle q_{1}\rangle+\cdots+\langle
q_{n}\rangle.$

$(\textbf{III.II})$, We suppose that $(\beta_{n}\alpha(p)-2\delta)_{+}$ is a
purely positive element, then By Theorem 2.3 (2), there is a non-zero positive
element $d$ orthogonal to $(\beta_{n}\alpha(p)-2\delta)_{+}$ such that

$(\beta_{n}\alpha(p)-3\delta)_{+}+d\lesssim(\beta_{n}\alpha(p)-2\delta)_{+}.$

With $G=\\{\gamma_{n}(a),\gamma_{n}(p_{i}),~{}\gamma_{n}(v_{i,j}^{k}):~{}1\leq
i\leq k+1,~{}1\leq j\leq k+1,0\leq k\leq n\\},$ and
$\varepsilon^{\prime\prime}>0$, $E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is
asymptotically tracially in $\mathcal{P}$, there exist a ${\rm C^{*}}$-algebra
$D$ in $\mathcal{P}$ and completely positive contractive linear maps
$\alpha^{\prime}:E\to D$ and $\beta_{n}^{\prime}:D\to E$, and
$\gamma_{n}^{\prime}:E\to E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1)^{\prime}$ the map $\alpha^{\prime\prime}$ is unital completely positive
linear map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{A})$ are
all projections, $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{A})=1_{A}$
for all $n\in\mathbb{N}$,

$(2)^{\prime}$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3)^{\prime}$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4)^{\prime}$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5)^{\prime}$ $\gamma_{n}^{\prime}\gamma_{n}(1)\lesssim d$ for all
$n\in\mathbb{N}$.

Since

$\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i})){\rm diag}(\gamma_{n}(q_{i})\otimes
1_{k},~{}0)(\gamma_{n}\otimes id_{M_{k}}({v}^{*}))-\gamma_{n}(p)\otimes
1_{k+1}\|<2\delta,$

for $0\leq i\leq n$, by $(1)^{\prime}$, we have

$\|\alpha^{\prime}\otimes id_{M_{k+1}}\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}){\rm diag}(\alpha^{\prime}\gamma_{n}(q_{i})\otimes
1_{k},~{}0)$ $\alpha^{\prime}\otimes id_{M_{k+1}}(\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}^{*}))-\alpha^{\prime}\gamma_{n}(p)\otimes
1_{k+1}\|<3\delta.$

By Theorem 2.3 (1), we have

$(k+1)\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq
k\langle(\alpha^{\prime}\gamma_{n}(q_{i})-\delta)_{+}\rangle.$

Since $D\in\Omega$, we have

$\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle.$

Since
$\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle,$
there exists $w\in M_{k+1}(D)$ such that

$\|w{\rm
diag}((\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+},(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+},\cdots,(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+})w^{*}-(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\|<\delta.$

By $(4)$, we have

$\|\beta_{n}^{\prime}\otimes id_{M_{k+1}}(w){\rm
diag}((\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+},(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+},\cdots,(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+})$
$\beta_{n}^{\prime}\otimes
id_{M_{k+1}}(w^{*})-(\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(p)-(n+10)\delta)_{+}\rangle\leq\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle.$

Therefore, we have

$\displaystyle\langle
p\rangle\leq\langle(\gamma_{n}(p)-(n+10)\delta)_{+})\rangle+\langle(\beta_{n}\alpha(p)-(n+10)\delta)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(p)\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-(n+12)\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p)-3\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(1_{A})\rangle$
$\displaystyle+\langle(\beta_{n}\alpha((a-\delta)_{+})-3\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-(n+8)\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle\leq\langle
d\rangle+\langle(\beta_{n}\alpha(p)-3\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-(n+10)\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}\alpha(q_{n})-\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}\alpha(q_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle q_{0}\rangle+\langle q_{1}\rangle+\cdots+\langle
q_{n}\rangle.$

$(\textbf{III.III})$, we suppose that $(\beta_{n}\alpha(p)-2\delta)_{+}$ is
Cuntz equivalent to a projection and there exist some $q_{i}$ such that
$\beta_{n}\alpha(q_{i})$ is not Cuntz equivalent to a projection. Choose a
projection $p_{0}$ such that$(\beta_{n}\alpha(p)-2\delta)_{+}$ is Cuntz
equivalent to $p_{0}.$ We may assume that
$(\beta_{n}\alpha(p)-2\delta)_{+}=p_{0}.$

We may also assume that $q_{0}$ such that $\beta_{n}\alpha(q_{0})$ is a pure
positive. By Theorem 2.3 (2), there exists a non-zero positive element $d$
orthogonal to $\beta_{n}\alpha(q_{0})$ such that
$\langle(\beta_{n}\alpha(q_{0})-\delta)_{+}+d\rangle\leq\langle\beta_{n}\alpha(q_{0})\rangle.$

With $G=\\{\gamma_{n}(a),\gamma_{n}(p_{i}),~{}\gamma_{n}(v_{i,j}^{k}):~{}1\leq
i\leq k+1,~{}1\leq j\leq k+1,0\leq k\leq n\\},$ and any sufficiently small
$\varepsilon^{\prime\prime}>0$, $E=\gamma_{n}(1)A\gamma_{n}(1)$, since $E$ is
asymptotically tracially in $\mathcal{P}$, there exist a ${\rm C^{*}}$-algebra
$D$ in $\mathcal{P}$ and completely positive contractive linear maps
$\alpha^{\prime}:E\to D$ and $\beta_{n}^{\prime}:D\to E$, and
$\gamma_{n}^{\prime}:E\to E\cap\beta_{n}^{\prime}(D)^{\perp}$ such that

$(1^{\prime})$ the map $\alpha^{\prime}$ is unital completely positive linear
map, $\beta_{n}^{\prime}(1_{D})$ and $\gamma_{n}^{\prime}(1_{A})$ are all
projections, $\beta_{n}^{\prime}(1_{D})+\gamma_{n}^{\prime}(1_{A})=1_{A}$ for
all $n\in\mathbb{N}$,

$(2^{\prime})$
$\|x-\gamma_{n}^{\prime}(x)-\beta_{n}^{\prime}(\alpha^{\prime}(x))\|<\varepsilon^{\prime\prime}$
for all $x\in G$ and for all $n\in{\mathbb{N}}$,

$(3^{\prime})$ $\alpha^{\prime}$ is an $F$-$\varepsilon^{\prime\prime}$
approximate embedding,

$(4^{\prime})$
$\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)-\beta_{n}^{\prime}(x)\beta_{n}(y)\|=0$
and $\lim_{n\to\infty}\|\beta_{n}^{\prime}(xy)\|=\|x\|$ for all $x,y\in D$,
and

$(5^{\prime})$ $\gamma_{n}^{\prime}(1)\gamma_{n}\lesssim d$ for all
$n\in\mathbb{N}$.

Since

$\|(\gamma_{n}\otimes id_{M_{k+1}}(v_{i})){\rm diag}(\gamma_{n}(q_{i})\otimes
1_{k},~{}0)(\gamma_{n}\otimes id_{M_{k}}({v}^{*}))-\gamma_{n}(p)\otimes
1_{k+1}\|<2\delta,$

for $0\leq i\leq n$, by $(1)^{\prime}$, we have

$\|\alpha^{\prime}\otimes id_{M_{k+1}}\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}){\rm diag}(\alpha^{\prime}\gamma_{n}(q_{i})\otimes
1_{k},~{}0)$ $\alpha^{\prime}\otimes id_{M_{k+1}}(\gamma_{n}\otimes
id_{M_{k+1}}(v_{i}^{*}))-\alpha^{\prime}\gamma_{n}(p)\otimes
1_{k+1}\|<3\delta.$

By Theorem 2.3 (1), we have

$(k+1)\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq
k\langle(\alpha^{\prime}\gamma_{n}(q_{i})-\delta)_{+}\rangle.$

Since $D\in\Omega$, we have

$\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle.$

Since
$\langle(\alpha^{\prime}\gamma_{n}(p)-(n+6)\delta)_{+}\rangle\leq\langle(\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle,$
there exists $w\in M_{k+1}(D)$ such that

$\|w{\rm
diag}((\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+},(\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+},\cdots,(\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+})w^{*}-(\alpha^{\prime}\gamma_{n}(p)-(n+3)\delta)_{+}\|<\delta.$

By $(4)$, we have

$\|\beta_{n}^{\prime}\otimes id_{M_{k+1}}(w){\rm
diag}((\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+},(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+},\cdots,(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+})$
$\beta_{n}^{\prime}\otimes
id_{M_{k+1}}(w^{*})-(\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(p)-(n+3)\delta)_{+}\|<2\delta.$

By Theorem 2.3 (1), we have

$\langle\beta_{n}^{\prime}(\alpha^{\prime}\gamma_{n}(p)-(n+10)\delta)_{+}\rangle\leq\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle.$

Therefore, we have

$\displaystyle\langle
p\rangle\leq\langle(\gamma_{n}(p)-(n+10)\delta)_{+})\rangle+\langle(\beta_{n}\alpha(p)-(n+10)\delta)_{+}\rangle$
$\displaystyle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(p)\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-(n+12)\delta)_{+}\rangle+\langle(\beta_{n}\alpha(p)-3\delta)_{+}\rangle\leq\langle\gamma_{n}^{\prime}\gamma_{n}(1_{A})\rangle$
$\displaystyle+\langle(\beta_{n}\alpha((a-\delta)_{+})-3\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}((a-(n+8)\delta)_{+})-2\delta)_{+}\rangle$
$\displaystyle\leq\langle
d\rangle+\langle(\beta_{n}\alpha(p)-3\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(p)-(n+10)\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}\alpha(q_{n})-\delta)_{+}\rangle$
$\displaystyle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}^{\prime}\alpha^{\prime}\gamma_{n}(q_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle(\beta_{n}\alpha(q_{0})-\delta)_{+}\rangle+\langle(\beta_{n}\alpha(q_{1})-\delta)_{+}\rangle+\cdots+\langle(\beta_{n}\alpha(q_{n})-\delta)_{+}\rangle$
$\displaystyle\leq\langle q_{0}\rangle+\langle q_{1}\rangle+\cdots+\langle
q_{n}\rangle.$

∎

######  Corollary 3.3.

Let $\mathcal{P}$ be a class of stably finite unital ${\rm C^{*}}$-algebras
such that for any $B\in\mathcal{P},$ ${\rm W}(B)$ is almost unperforated. Then
${\rm W}(A)$ is almost unperforated for any simple unital ${\rm
C^{*}}$-algebra $A\in{\rm TA}\mathcal{P}$.

###### Proof.

This is a special case of Theorem 3.1 and also of Theorem 3.2. ∎

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