# certain tracially nuclear dimensional for certain crossed product ${\rm
C^{*}}$-algebras

Qingzhai Fan and Jiahui Wang  Qingzhai Fan
Department of Mathematics
Shanghai Maritime University
Shanghai
China
201306  Jiahui Wang
Department of Mathematics
Shanghai Maritime University
Shanghai
China
201306<EMAIL_ADDRESS>
###### Abstract.

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras which have the second
type tracial nuclear dimensional at moat $n$ (or have tracial nuclear
dimensional at most $n$). Let $A$ be an infinite dimensional unital simple
${\rm C^{*}}$-algebra such that $A$ is asymptotical tracially in $\Omega$.
Then ${\rm T^{2}dim_{nuc}}(A)\leq n$ (or ${\rm Tdim_{nuc}}(A)\leq n$). As an
application, let $A$ be an infinite dimensional simple separable amenable
unital ${\rm C^{*}}$-algebra with ${\rm T^{2}dim_{nuc}}(A)\leq n$ (or ${\rm
Tdim_{nuc}}(A)\leq n$). Suppose that $\alpha:G\to{\rm Aut}(A)$ is an action of
a finite group $G$ on $A$ which has the tracial Rokhlin property. Then ${\rm
T^{2}dim_{nuc}}({{\rm C^{*}}(G,A,\alpha)})\leq n$ (or ${\rm Tdim_{nuc}}$
$({{\rm C^{*}}(G,A,\alpha)})\leq n$).

Key words ${\rm C^{*}}$-algebras, tracial approximation, tracially
$\mathcal{Z}$-absorbing.

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

##  1\. Introduction

The Elliott program for the classification of amenable ${\rm C^{*}}$-algebras
might be said to have begun with the ${\rm K}$-theoretical classification of
AF algebras in [5]. Since then, many classes of ${\rm C^{*}}$-algebras have
been classified by the Elliott invariant. A major next step was the
classification of simple $\rm AH$ algebras without dimension growth (in the
real rank zero case see [7], and in the general case see [8]). A crucial
intermediate step was Lin’s axiomatization of Elliott-Gong’s decomposition
theorem for simple $\rm AH$ algebras of real rank zero (classified by Elliott-
Gong in [7]) and Gong’s decomposition theorem ([19]) for simple $\rm AH$
algebras (classified by Elliott-Gong-Li in [8]). Lin introduced the concepts
of $\rm TAF$ and $\rm TAI$ ([32] and [33]). Instead of assuming inductive
limit structure, Lin started with a certain abstract (tracial) approximation
property. This led eventually to the classification of simple separable
amenable stably finite ${\rm C^{*}}$-algebras with finite nuclear dimension in
the UCT class (see [22], [23], [11], [44]).

Inspired by Lin’s tracial interval algebras in [33], Elliott and Niu in [12]
considered the natural notion of tracial approximation by other classes of
${\rm C^{*}}$-algebras. Let $\Omega$ be a class of unital ${\rm
C^{*}}$-algebras. Then the class of simple separable ${\rm C^{*}}$-algebras
which can be tracially approximated by ${\rm C^{*}}$-algebras in $\Omega,$
denoted by ${\rm TA}\Omega$, is defined as follows. A simple unital ${\rm
C^{*}}$-algebra $A$ is said to belong to the class ${\rm TA}\Omega$ if, for
any $\varepsilon>0,$ any finite subset $F\subseteq A,$ and any non-zero
element $a\geq 0,$ there are a projection $p\in A$ and a ${\rm
C^{*}}$-subalgebra $B$ of $A$ with $1_{B}=p$ and $B\in\Omega$ such that

$(1)$ $\|xp-px\|<\varepsilon$, for all $x\in F$,

$(2)$ $pxp\in_{\varepsilon}B$, for all $x\in F$, and

$(3)$ $1-p$ is Murray-von Neumann equivalent to a projection in
$\overline{aAa}$.

The Rokhlin property in ergodic theory was adapted to the context of von
Neumann algebras by Connes in [4]. It was adapted by Herman and Ocneanu for
automorphisms of UHF algebras in [25]. Izumi [29], Rørdam [41] and Kishimoto
[31] considered the Rokhlin property in a much more general ${\rm
C^{*}}$-algebra context. More recently, Phillips and Osaka studied actions of
a finite group and of the group $\mathbb{Z}$ of integers on certain simple
${\rm C^{*}}$-algebras which have a modified Rokhlin property in [35] and
[39].

In [39], Phillips show that let $A$ be an infinite-dimensional simple
separable unital ${\rm C^{*}}$-algebra with tracial topological rank zero. Let
$\alpha:G\to{\rm Aut}(A)$ is an action of a finite group $G$ on $A$ which has
the tracial Rokhlin property. Then ${\rm C^{*}}(G,A,\alpha)$ has tracial
topological rank zero.

Inspired by Phillips’s above result, in [16], Fan and Fang showed the
following result:

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras such that $\Omega$ is
closed under passing to unital hereditary ${\rm C^{*}}$-subalgebra and
tensoring matrix algebras. Let $A\in$ $\rm{TA}\Omega$ (tracial approximated by
${\rm C^{*}}$-algebras in $\Omega$) be an infinite-dimensional simple unital
${\rm C^{*}}$-algebra. Suppose that $\alpha:G\to{\rm{Aut}}(A)$ is an action of
a finite group $G$ on $A$ which has tracial Rokhlin property. Then the crossed
product ${\rm C^{*}}$-algebra ${\rm C^{*}}(G,A,\alpha)$ belongs to
$\rm{TA}\Omega$.

Note that: a similar result was obtained independently using different method
by H. H. Lee and H. Osaka in [24].

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras such that for any
$B\in\Omega$, $B$ has property $H$ (we assume that property $H$ is closed
under tensoring with matrix algebras and under passing to unital hereditary
${\rm C^{*}}$-subalgebras). If we show that $A$ has the property $H$ for any
unital simple separable $A\in\rm TA\Omega$, then by the above result, we can
get the following result:

Let $A$ be a unital ${\rm C^{*}}$-algebras such that $A$ has property $H$.
Suppose that $\alpha:G\to$ ${\rm{Aut}}(A)$ is an action of a finite group $G$
on $A$ which has the tracial Rokhlin property. Then the crossed product ${\rm
C^{*}}$-algebra ${\rm C^{*}}(G,A,\alpha)$ also has property $H$.

In the quest to classify simple separable nuclear ${\rm C^{*}}$-algebras, as
suggested by G. A. Elliott, it has become necessary to invoke some regularity
property of the ${\rm C^{*}}$-algebras. There are three regularity properties
of particular interest: tensorial absorption of the Jiang-Su algebra
$\mathcal{Z}$, also called $\mathcal{Z}$-stability; finite nuclear dimension;
and strict comparison of positive elements. The latter can be reformulated as
an algebraic property of Cuntz semigroup, called almost unperforated. Winter
and Toms have conjectured that these three fundamental properties are
equivalent for all separable, simple, nuclear ${\rm C^{*}}$-algebras.

Hirshberg and Oroviz introduced tracially $\mathcal{Z}$-stable in [26], they
showed that $\mathcal{Z}$-stable is equivalent to tracially
$\mathcal{Z}$-stable for unital simple separable amenable ${\rm
C^{*}}$-algebra in [26].

Inspired by Hirshberg and Oroviz’s tracially $\mathcal{Z}$-stable, and
inspired by the work of Elliott, Gong, Lin, and Niu in [9], [20] and [21], and
also in order to search a tracial version of Toms-Winter conjecture, Fu and
Lin introduced some class of tracial nuclear dimensional ${\rm
C^{*}}$-algebras in [17] and [18]. Let $\Omega$ be a class of ${\rm
C^{*}}$-algebras. Fu and Lin introduced a class of ${\rm C^{*}}$-algebras
which was called asymptotical tracially in $\Omega$, and denoted this class of
${\rm C^{*}}$-algebras by $\rm AT\Omega$ (see Definition 2.7).

In this paper, we will show the following results:

Let $\Omega$ be a class of unital separable $\rm{C^{*}}$-algebras and let $A$
be a unital simple separable infinite dimensional amenable
$\rm{C^{*}}$-algebra. Then $A\in\rm{AT}\Omega$ if and only if
$A\in\rm{TA(AT\Omega)}$ (see Definition 2.7 and Definition 2.2).

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras which have the second
type tracial nuclear dimensional at moat $n$ (or have tracial nuclear
dimensional at most $n$). Let $A$ be a unital simple ${\rm C^{*}}$-algebra
such that $A$ is asymptotical tracially in $\Omega$. Then ${\rm
T^{2}dim_{nuc}}(A)\leq n$ (or ${\rm Tdim_{nuc}}(A)\leq n$) (see Definition 2.9
and Definition 2.7).

As applications, we can get the following results:

Let $A$ be an infinite dimensional simple separable amenable unital ${\rm
C^{*}}$-algebra such that ${\rm T^{2}dim_{nuc}}(A)\leq n$. Suppose that
$\alpha:G\to{\rm Aut}(A)$ is an action of a finite group $G$ on $A$ which has
the tracial Rokhlin property. Then ${\rm T^{2}dim_{nuc}}({{\rm
C^{*}}(G,A,\alpha)})\leq n$.

Let $A$ be an infinite dimensional simple separable amenable unital ${\rm
C^{*}}$-algebra such that ${\rm Tdim_{nuc}}(A)\leq n$. Suppose that
$\alpha:G\to{\rm Aut}(A)$ is an action of a finite group $G$ on $A$ which has
the tracial Rokhlin property. Then ${\rm Tdim_{nuc}}({{\rm
C^{*}}(G,A,\alpha)})\leq n$ (Note that: this result also can be obtained by
Theorem 9.3 of [18] and Theorem 3.1 of [6] and Theorem 3.3).

##  2\. Preliminaries and definitions

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 $a\precsim b$) 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 $a\precsim
b$ and $b\precsim a$. 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 $A$. ${\rm W}(A)$ 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

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

Let $A$ be a unital ${\rm C^{*}}$-algebra. Recall that a positive element
$a\in A$ is called purely positive if $a$ is not Cuntz equivalent to a
projection. Let $A$ be a stably finite ${\rm C^{*}}$-algebra and let $a\in A$
be a positive element. Then $a$ is a purely positive element or $a$ is Cuntz
equivalent to a projection.

Given $a$ in $A_{+}$ and $\varepsilon>0,$ we denote by $(a-\varepsilon)_{+}$
the element of ${\rm C^{*}}(a)$ corresponding (via the functional calculus) to
the function $f(t)={\max(0,t-\varepsilon)},$ $t\in\sigma(a)$. By the
functional calculus, it follows in a straightforward manner that
$((a-\varepsilon_{1})_{+}-\varepsilon_{2})_{+}=(a-(\varepsilon_{1}+\varepsilon_{2}))_{+}.$

The following facts are well known.

######  Theorem 2.1.

([3], [26], [40], [42].) Let $A$ be a ${\rm C^{*}}$-algebra.

$(1)$ Let $a,~{}b\in A_{+}$ and any $\varepsilon>0$ be such that
$\|a-b\|<\varepsilon$. Then there is a contraction $d$ in $A$ with
$(a-\varepsilon)_{+}=dbd^{*}$.

$(2)$ Let $a$ be a positive element of $A$ not Cuntz equivalent to a
projection. Let $\delta>0$, and let $f\in C_{0}(0,1]$ be a non-negative
function with $f=0$ on $(\delta,1),$ $f>0$ on $(0,\delta)$, and $\|f\|=1$.
Then $f(a)\neq 0$ and $(a-\delta)_{+}+f(a)\precsim a.$

$(3)$ Let $a,~{}b\in A$ satisfy $0\leq a\leq b$. Let $\varepsilon\geq 0$. Then
$(a-\varepsilon)_{+}\precsim(b-\varepsilon)_{+}$ (Lemma 1.7 of [40]).

Inspired by Lin’s tracial approximation by interval algebras in [33], Elliott
and Niu in [12] considered the natural notion of tracial approximation by
other classes of ${\rm C^{*}}$-algebras.

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras. Then the class of
simple separable ${\rm C^{*}}$-algebras which can be tracially approximated by
${\rm C^{*}}$-algebras in $\Omega,$ denoted by ${\rm TA}\Omega$, is defined as
follows:

######  Definition 2.2.

([12], [33].) A simple unital ${\rm C^{*}}$-algebra $A$ is said to belong to
the class ${\rm TA}\Omega$ if, for any $\varepsilon>0,$ any finite subset
$\mathcal{G}\subseteq A,$ and any non-zero element $a\geq 0,$ there are a
projection $p\in A$ and a ${\rm C^{*}}$-subalgebra $B$ of $A$ with $1_{B}=p$
and $B\in\Omega$ such that

$(1)$ $\|xp-px\|<\varepsilon$, for all $x\in\mathcal{G}$,

$(2)$ $pxp\in_{\varepsilon}B$, for all $x\in\mathcal{G}$, and

$(3)$ $1-p\precsim a$.

Let $a$ and $b$ be two positive elements in a $\rm C^{*}$-algebra $A$. We
write $[a]\leq[b]$ (cf Definition 3.5.2 in [34]), if there exists a partial
isometry $v\in A^{**}$ such that, for every $c\in{\rm
Her}(a),~{}v^{*}c,~{}cv\in A,~{}vv^{*}=P_{a}$, where $P_{a}$ is the range
projection of $a$ in $A^{**}$, and $v^{*}cv\in{\rm Her}(b)$. We write
$[a]=[b]$ if $v^{*}{\rm Her}(a)v={\rm Her}(b)$. Let $n$ be a positive integer.
We write $n[a]\leq[b]$, if there are $n$ mutually orthogonal positive elements
$b_{1},~{}b_{2},~{}\cdots,~{}b_{n}\in{\rm Her}(b)$ such that $[a]\leq[b_{i}],$
for $i=1,~{}2,~{}\cdots,~{}n$. (cf. Definition 1.1 in [38], Definition 3.2 in
[37], or Definition 3.5.2 in [34].)

Let $0<\sigma_{1}<\sigma_{2}\leq 1$ be two positive numbers. Define

$f_{\sigma_{1}}^{\sigma_{2}}(t)=\left\\{\begin{array}[]{ll}1&if\
t\geq\sigma_{2}\\\ \frac{t-\sigma_{1}}{\sigma_{2}-\sigma_{1}}&if\
\sigma_{1}\leq t\leq\sigma_{2}\\\ 0&if\ 0<t\leq\sigma_{1}.\end{array}\right.$

######  Definition 2.3.

([27])  Let $\Omega$ be a class of unital $\rm C^{*}$-algebras. A unital $\rm
C^{*}$-algebra $A$ is said to have property ${\rm(I)}$ if, for any positive
numbers $0<\sigma_{3}<\sigma_{4}<\sigma_{1}<\sigma_{2}<1$, any
$\varepsilon>0$, any finite subset ${F}\subseteq A$, any finite nonzero
positive element $G\subseteq A_{+}$, and any integer $n>0$, there exist a
nonzero projection $p\in A$ and a $C^{*}$-subalgebra $B$ of $A$ with
$B\in\Omega$ and $1_{B}=p$, such that

$(1)$ $\|xp-px\|<\varepsilon$, for all $x\in{F}$,

$(2)$ $pxp\in_{\varepsilon}B$, for all $x\in{F}$,

$(3)$
$n[f^{\sigma_{2}}_{\sigma_{1}}((1-p)a(1-p))]\leq[f^{\sigma_{4}}_{\sigma_{3}}(pap)]$,
for any $a\in G$.

######  Theorem 2.4.

( [27])  Let $A$ be a unital simple $\rm{C^{*}}$-algebra. Then $A$ belong to
${\rm TA}\Omega$ if and only if $A$ has property ${\rm(I)}$.

######  Theorem 2.5.

Let $\Omega$ be a class of simple unital $\rm C^{*}$-algebra. Let $A$ be a
unital simple $\rm C^{*}$-algebra. Then $A\in\rm{TA}\Omega$, if and only if,
for any $\varepsilon>0$, any finite $F\subseteq A$, and any nonzero positive
element $a\in A_{+}$, there exist a nonzero projection $p\in A$ and a $\rm
C^{*}$-subalgebra $B$ of $A$ with $1_{B}=p$ and $B\in\Omega$, such that

$(1)$ $\|px-xp\|<\varepsilon$, for all $x\in F$,

$(2)$ $pxp\in_{\varepsilon}B$, for all $x\in F$,

$(3)$ $1-p\precsim a$, and

$(4)$ $\|pxp\|\geq\|x\|-\varepsilon$, for all $x\in F$.

###### Proof.

$\Longleftarrow$: This is obvious.

$\Longrightarrow$: We must show that for any $\varepsilon>0$, any finite
$F=\\{x_{1},~{}\cdots,~{}x_{n}\\}\subseteq A$ (we may assume that
$\|x_{i}\|=1$, for any $1\leq i\leq n$), any nonzero positive element $a\in
A_{+}$, there exist a nonzero projection $p\in A$ and a $\rm C^{*}$-subalgebra
$B$ of $A$ with $1_{B}=p$, and $B\in\Omega$, such that

$(1)$ $\|px-xp\|<\varepsilon$, for all $x\in F$,

$(2)$ $pxp\in_{\varepsilon}B$, for all $x\in F$,

$(3)$ $1-p\precsim a$, and

$(4)$ $\|pxp\|\geq\|x\|-\varepsilon$, for all $x\in F$.

Since $A\in\rm{TA}\Omega$, with $\varepsilon>0$, and
$G=\\{x_{1},~{}\cdots,~{}x_{n},~{}x_{1}^{*}x_{1},~{}\cdots,$
$~{}x_{n}^{*}x_{n}\\}\subseteq A$, there exist a nonzero projection $p\in A$,
and a $\rm C^{*}$-subalgebra $B$ of $A$ with $1_{B}=p$, and $B\in\Omega$, such
that

$(1^{\prime})$ $\|px-xp\|<\varepsilon/3$, for all $x\in G$,

$(2^{\prime})$ $pxp\in_{\varepsilon/3}B$, for all $x\in G$, and

$(3^{\prime})$ $1-p\precsim a$.

By Theorem 2.4, for $x_{i}^{*}x_{i}\in A_{+}$, with $\|x_{i}^{*}x_{i}\|=1$
($1\leq i\leq n$), let
$0<\sigma_{3}=1-4\varepsilon<\sigma_{4}=1-3\varepsilon<\sigma_{1}=1-\varepsilon<\sigma_{2}<1$,
we have

$(4^{\prime})$
$[f^{\sigma_{2}}_{\sigma_{1}}((1-p)x_{i}^{*}x_{i}(1-p))]\leq[f^{\sigma_{4}}_{\sigma_{3}}(px_{i}^{*}x_{i}p)]$,
for $1\leq i\leq n$.

If $\|px_{i}^{*}x_{i}p\|<1-\varepsilon$, then we have
$f^{\sigma_{4}}_{\sigma_{3}}(px_{i}^{*}x_{i}p)=0$, by $(4^{\prime})$, one has

$f^{\sigma_{2}}_{\sigma_{1}}((1-p)x_{i}^{*}x_{i}(1-p))=0.$

So we has

$\|(1-p)x_{i}^{*}x_{i}(1-p)\|<1-4\varepsilon.$

By $(1^{\prime})$ and $(2^{\prime})$, for $i=1,~{}\cdots,~{}n$, we have

$\|x_{i}^{*}x_{i}-(1-p)x_{i}^{*}x_{i}(1-p)-px_{i}^{*}x_{i}p\|$

$\leq\|x_{i}^{*}x_{i}-(1-p)x_{i}^{*}x_{i}-px_{1}^{*}x_{i}\|$

$+\|(1-p)x_{i}^{*}x_{i}-(1-p)x_{i}^{*}x_{i}(1-p)\|+\|px_{i}^{*}x_{i}-px_{i}^{*}x_{i}p\|$

$<\varepsilon/3+\varepsilon/3+\varepsilon/3=\varepsilon$.

Then, we have

$1=\|x_{i}^{*}x_{i}\|\leq\|(1-p)x_{i}^{*}x_{i}(1-p)+px_{i}^{*}x_{i}p\|+\varepsilon$

$=\max\\{\|(1-p)x_{i}^{*}x_{i}(1-p),~{}\|px_{i}^{*}x_{i}p\|\\}+\varepsilon<1$,
this is a contradictory.

Therefore, for any $1\leq i\leq n$, we have

$\|px_{i}^{*}x_{i}p\|\geq 1-\varepsilon.$

Since $\|px_{i}p\|\leq\|x\|=1$, then, for any $1\leq i\leq n$, we have

$\|px_{i}p\|\geq\|px_{i}p\|^{2}\geq\|px_{i}^{*}x_{i}p\|-\varepsilon\geq
1-2\varepsilon.$

With $2\varepsilon$ in place of $\varepsilon$. ∎

Winter and Zacharias introduce nuclear dimension for ${\rm C^{*}}$-algebras in
[45].

######  Definition 2.6.

([45].) Let $A$ be a ${\rm C^{*}}$-algebra, $m\in{\mathbb{N}}$. A complete
positive compression $\varphi:F\to A$ is $m$-decomposable (where $F$ is finite
dimensional ${\rm C^{*}}$-algebra ), if there is a decomposition

$F=F^{(0)}\oplus F^{(1)}\oplus\cdots\oplus F^{(m)}$

such that the restriction $\varphi^{(i)}$ of $\varphi$ to $F^{(i)}$ has order
zero (A complete positive map $\varphi:A\to B$ is order zero if for any
positive elements $a,b$ with $ab=0$ we have $\varphi(a)\varphi(b)=0$) for each
$i\in\\{0,\cdots,m\\}$, we say $\varphi$ is $m$-decomposable with respect to
$F=F^{(0)}\oplus F^{(1)}\oplus\cdots\oplus F^{(m)}$.

A has nuclear dimension $m$, write ${\rm{dim_{nuc}}}(A)=m$, if $m$ is the
least integer such that the following holds: For any finite subset
$\mathcal{G}\subseteq A$ and $\varepsilon>0$, there is a finite dimension
complete positive compression approximation $(F,\varphi,\psi)$ for
$\mathcal{G}$ to within $\varepsilon$ (i.e., $F$ is finite dimensional
$\psi:A\to F$ and $\varphi:F\to A$ are complete positive and
$\|\varphi\psi(b)-b\|<\varepsilon$ for any $b\in\mathcal{G}$) such that $\psi$
is complete positive compression, and $\varphi$ is $m$-decomposable with
complete positive compression order zero components $\varphi^{i}$. If no such
$m$ exists, we write $dim_{nuc}(A)=\infty$.

Inspired by Hirshberg and Orovitz’s tracial $\mathcal{Z}$-absorption in [26],
Fu introduced some notion of tracial nuclear dimension in his doctoral
dissertation [17] (see also [18]).

######  Definition 2.7.

([18]) Let $A$ be a unital simple $\rm C^{*}$-algebra. Let $n\in\mathbb{N}$ be
an integer. $A$ is said to have tracial nuclear dimension at most $n$, and
denoted by ${\rm Tdim_{nuc}}A\leq n$, if for any finite subset
$\mathcal{F}\subseteq A$, for any $\varepsilon>0$ and for any nonzero positive
element $a\in A_{+}$, there exist a finite dimensional $\rm C^{*}$-algebra
$F$, a completely positive contractive map $\alpha:A\rightarrow F$, a nonzero
piecewise contractive $n$-decomposable completely positive map
$\beta:F\rightarrow A$, and a completely positive contractive map
$\gamma:A\rightarrow A\cap\beta^{\perp}(F)$ such that

$(1)$ $\|x-\gamma(x)-\beta\alpha(x)\|<\varepsilon$, for all $x\in\mathcal{F}$,
and

$(2)$ $\gamma(1_{A})\precsim a$.

Note that: Tracial nuclear dimension at most $n$ is closed under tensoring
with matrix algebras and under passing to unital hereditary $\rm
C^{*}$-subalgebras.

######  Theorem 2.8.

Let $A$ be a unital simple ${\rm C^{*}}$-algebra. Then ${\rm Tdim_{nuc}}A\leq
n$, if and only if, if for any finite subset $\mathcal{F}\subseteq A$, for any
$\varepsilon>0$ and for any nonzero positive element $a\in A_{+}$, there exist
a finite dimensional $\rm C^{*}$-algebra $F$, a completely positive
contractive map $\alpha:A\rightarrow F$, a nonzero piecewise contractive
$n$-decomposable completely positive map $\beta:F\rightarrow A$, and a
completely positive contractive map $\gamma:A\rightarrow
A\cap\beta^{\perp}(F)$ such that

$(1)$ $\|x-\gamma(x)-\beta\alpha(x)\|<\varepsilon$, for all $x\in\mathcal{F}$,
and

$(2)$ $(\gamma(1_{A})-\varepsilon)_{+}\precsim a$.

###### Proof.

As in the proof of Theorem 2.12, we can arrange that $\gamma_{n}(1_{A})$ is a
projection. Then, we have
$(\gamma_{n}(1_{A})-\varepsilon)_{+}\thicksim\gamma_{n}(1_{A})$. ∎

######  Definition 2.9.

([17]) Let $A$ be a $\rm C^{*}$-algebra. Let $n\in\mathbb{N}$ be an integer.
$A$ is said to have the second type tracial nuclear dimension at most $n$, and
denoted by ${\rm T^{2}dim_{nuc}}A\leq n$, if for any finite positive subset
$\mathcal{F}\subseteq A$, for any $\varepsilon>0$ and for any nonzero positive
element $a\in A_{+}$, there exist a finite dimensional $\rm C^{*}$-algebra
$F=F_{0}\oplus\cdots\oplus F_{n}$ and completely positive maps
$\psi:A\rightarrow F$, $\varphi:F\rightarrow A$ such that

$(1)$ for any $x\in F$, there exists $x^{\prime}\in A_{+}$, such that
$x^{\prime}\precsim a$, and $\|x-x^{\prime}-\varphi\psi(x)\|<\varepsilon$,

$(2)$ $\|\psi\|\leq 1$, and

$(3)$ $\varphi|_{F_{i}}$ is a completely positive contractive order zero map,
for any $0\leq i\leq n$.

Let $A$ be a unital $\rm C^{*}$-algebra. It is easy to know that
${\rm{Tdim_{nuc}}}(A)\leq n$ implies that ${\rm{T^{2}dim_{nuc}}}(A)\leq n$.

Note that: Second type tracial nuclear dimension at most $n$ is closed under
tensoring with matrix algebras and under passing to unital hereditary $\rm
C^{*}$-subalgebras.

The tracial Rokhlin property was introduced by Phillips in [39] for finite
group actions.

######  Definition 2.10.

([39].) Let $A$ be a separable simple unital ${\rm C^{*}}$-algebra, and let
$\alpha:G\to{\rm Aut}(A)$ be an action of a finite group $G$ on $A$. We say
that $\alpha$ has the tracial Rokhlin property if, for any finite set
$\mathcal{G}\subseteq A,$ any $\varepsilon>0,$ and any non-zero positive
element $b\in A,$ there are mutually orthogonal projections $e_{g}\in A$ for
$g\in G$ such that

$(1)$ $\|{\alpha_{g}}(e_{h})-e_{gh}\|<\varepsilon$, for all $g,~{}h\in G,$

$(2)$ $\|e_{g}d-de_{g}\|<\varepsilon$, for all $g\in G$, and all
$d\in{\mathcal{G}},$

$(3)$ with $e=\Sigma_{g\in G}e_{g},$ the projection $1-e$ is Murray-von
Neumann equivalent to a projection in the hereditary ${\rm C^{*}}$-subalgebra
of $A$ generated by $b$, and

$(4)$ $\|ebe\|\geq\|b\|-\varepsilon$.

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras. Then the class of
${\rm C^{*}}$-algebras which are asymptotical tracially in $\Omega$, denoted
by ${\rm AT}\Omega$ was introduced by Fu and Lin in [18].

######  Definition 2.11.

([18]) Let $A$ be a unital simple $\rm C^{*}$-algebra and let $\Omega$ be a
class of $\rm C^{*}$-algebras. We say $A$ is asymptotical tracially in
$\Omega$, if for any finite subset $\mathcal{F}\subseteq A$, for any
$\varepsilon>0$ and for any nonzero positive element $a\in A_{+}$, there exist
a $\rm C^{*}$-algebra $B$ in $\Omega$, completely positive contractive maps
$\alpha:A\rightarrow B$, $\beta_{n}:B\rightarrow A$, and
$\gamma_{n}:A\rightarrow A\cap\beta_{n}^{\perp}(B)(n\in\mathbb{N})$ such that

$(1)$ $\|x-\gamma_{n}(x)-\beta_{n}\alpha(x)\|<\varepsilon$, for all
$x\in\mathcal{F}$, and for all $n\in{\mathbb{N}}$,

$(2)$ $\alpha$ is a $(\mathcal{F},\varepsilon)$-approximate embedding,

$(3)$
$\mathop{\lim}\limits_{n\rightarrow\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$
and $\mathop{\lim}\limits_{n\rightarrow\infty}\|\beta_{n}(x)\|=\|x\|$, for all
$x,~{}y\in B$, and

$(4)$ $\gamma_{n}(1_{A})\precsim a$, for all $n\in\mathbb{N}$.

######  Theorem 2.12.

([18]) Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras such that
$\Omega$ is closed under tensoring with matrix algebras and under passing to
unital hereditary ${\rm C^{*}}$-subalgebras. Let $A$ be a unital simple ${\rm
C^{*}}$-algebra. Then $A$ is asymptotical tracially in $\Omega$ (i.e.
$A\in{\rm AT}\Omega$), if and only if, for any $\varepsilon>0,$ any finite
subset $\mathcal{G}\subseteq A,$ and any non-zero element $a\geq 0,$ there
exist a ${\rm C^{*}}$-algebra $D$ in $\Omega$ and a nonzero completely
positive contractive linear map $\alpha:A\to D$ and completely positive
contractive linear maps $\beta_{n}:D\to A$, $\gamma_{n}:A\to
A\cap\beta_{n}(D)^{\perp}$ such that

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

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

$(3)$ $\alpha$ is a $\mathcal{G}$-$2\varepsilon$ approximate embedding,

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

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

######  Theorem 2.13.

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras such that $\Omega$ is
closed under tensoring with matrix algebras and under passing to unital
hereditary ${\rm C^{*}}$-subalgebras. Let $A$ be a unital simple ${\rm
C^{*}}$-algebra. Then $A$ is asymptotical tracially in $\Omega$ (i.e.
$A\in{\rm AT}\Omega$), if and only if, for any finite subset
$\mathcal{F}\subset A$, for any $\varepsilon>0$ and for any nonzero positive
element $a\in A_{+}$, there exist a $\rm C^{*}$-algebra $B$ in $\Omega$,
completely positive contractive maps $\alpha:A\rightarrow B$,
$\beta_{n}:B\rightarrow A$, and $\gamma_{n}:A\rightarrow
A\cap\beta_{n}^{\perp}(B)(n\in\mathbb{N})$ such that

$(1)$ $\|x-{\varepsilon}\gamma_{n}(x)-\beta_{n}\alpha(x)\|<\varepsilon$, for
all $x\in\mathcal{F}$, and for all $n\in{\mathbb{N}}$,

$(2)$ $\alpha$ is a $(\mathcal{F},2\varepsilon)$-approximate embedding,

$(3)$
$\mathop{\lim}\limits_{n\rightarrow\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$
and $\mathop{\lim}\limits_{n\rightarrow\infty}\|\beta_{n}(x)\|=\|x\|$, for all
$x,~{}y\in B$,and

$(4)$ $(\gamma_{n}(1_{A})-\varepsilon)_{+}\precsim a$, for all
$n\in\mathbb{N}$.

###### Proof.

By Theorem 2.12, we can arrange that $\gamma_{n}(1_{A})$ is a projection.
Then, we have $(\gamma_{n}(1_{A})-\varepsilon)_{+}\thicksim\gamma_{n}(1_{A})$.
∎

######  Lemma 2.14.

([18].) If the class $\Omega$ is closed under tensoring with matrix algebras
and under passing to unital hereditary ${\rm C^{*}}$-subalgebras, then the
class of asymptotically tracially in $\Omega$ is closed under tensoring with
matrix algebras and under passing to unital hereditary ${\rm
C^{*}}$-subalgebras.

The following theorem is Theorem 2.3.13 in [34].

######  Theorem 2.15.

([34].) Let $A$ be ${\rm C^{*}}$-algebra, $B$ be a ${\rm C^{*}}$-subalgebra of
$A$ and $C$ be another ${\rm C^{*}}$-algebra. Suppose that $\phi:B\to C$ is a
contractive completely positive linear map. If either $B$ or $C$ is amenable,
the for any finite subset $\mathcal{G}\subset B$ and $\varepsilon>0$, there is
a contractive completely positive linear map $\widetilde{\phi}:A\to C$ such
that

$\|\widetilde{\phi}-\phi\|<\varepsilon~{}~{}\rm{on}~{}\mathcal{G}.$

######  Definition 2.16.

Let $A$ and $B$ be ${\rm C^{*}}$-algebra, let $\varphi:A\to B$ be a map, let
$\mathcal{G}\subset A$, and let $\varepsilon>0$. The map $\varphi$ is called
$\mathcal{G}$-$\varepsilon$-multiplicative, or called
$\varepsilon$-multiplicative on $\mathcal{G}$, if for any $x,y\in F$,
$\|\varphi(xy)-\varphi(x)\varphi(y)\|<\varepsilon$. If, in addition, for any
$x\in\mathcal{G}$, $|\|\varphi(x)\|-\|x\||<\varepsilon$, then we say $\varphi$
is an $\mathcal{G}$-$\varepsilon$-approximate embedding.

######  Definition 2.17.

([18].) Let $A$ and $B$ be ${\rm C^{*}}$-algebra and let $\varphi:A\to B$ be a
map. let $\varepsilon>0$. If, for any $a_{1},a_{2}\in A_{+}^{1}$ with
$a_{1}a_{2}=0$, we have $\|\varphi(a_{1})\varphi(a_{2})\|\leq\varepsilon$,
then we say $\varphi$ is an $\varepsilon$-almost order zero map.

######  Definition 2.18.

([18].) Let $A$ be a ${\rm C^{*}}$-algebra and let $F$ be a finite dimensional
${\rm C^{*}}$-algebra. Let $\varphi:F\to A$ be an integer. The map $\varphi$
is called $(n-\varepsilon)$-dividable if $F$ can be written as $F=F_{0}\oplus
F_{1}\oplus\cdots\oplus F_{n}$ (where $F_{i}$ are ideals of $F$) such that
$\varphi|F_{i}$ is a completely

######  Theorem 2.19.

([18].) For any finite dimension $\rm C^{*}$-algebra $F$ and any
$\varepsilon>0$,there exists $\delta>0$ such that, for any $\rm C^{*}$-algebra
$A$ and any completely positive contractive map $\varphi:F\rightarrow A$ which
is $\delta$-almost order zero, there exist a completely positive contractive
order zero map $\psi:F\rightarrow A$ satisfying
$\|\varphi-\psi\|\leq\varepsilon$.

##  3\. the main result

######  Theorem 3.1.

Let $\Omega$ be a class of unital $\rm{C^{*}}$-algebras. Then $A$ is
asymptotically tracially in $\Omega$ for any unital simple separable infinite
dimensional amenable $\rm{C^{*}}$-algebra $A\in{{\rm TA}}({\rm AT}\Omega)$.

###### Proof.

We need to show that for any $\varepsilon>0$, any finite subset
$\mathcal{G}=\\{a_{1},a_{2},$ $\cdots,a_{k}\\}\subseteq A$ (we may assume that
$\|a_{i}\|\leq 1$ for all $1\leq i\leq k$), and any nonzero positive element
$b$, there exist a ${\rm C^{*}}$-algebra $B$ in $\Omega$ and completely
positive contractive linear maps $\alpha:A\to B$ and $\beta_{n}:B\to A$, and
$\gamma_{n}:A\to A$ such that

$(1)$ $\|x-\gamma_{n}(x)-\beta_{n}\alpha(x)\|<\varepsilon$, for all
$x\in\mathcal{G}$, and for all $n\in{\mathbb{N}}$,

$(2)$ $\alpha$ is a $\mathcal{G}$-$\varepsilon$ approximate embedding,

$(3)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(x)\|=\|x\|$, for all $x,~{}y\in B$, and

$(4)$ $\gamma_{n}(1)\precsim a$, for all $n\in\mathbb{N}$.

Since $A$ is an infinite-dimensional simple unital $\rm{C^{*}}$-algebra there
exist nonzero positive elements $b_{1},~{}b_{2}\in A_{+}$, such that
$b_{1}b_{2}=0$ and $b_{1}+b_{2}\precsim b$.

Since $A\in{{\rm TA}}({\rm AT}\Omega)$, by Theorem 2.5, for any $\delta>0$
(with $\delta<\varepsilon$), any finite subset
$\mathcal{H}=\\{a_{1},~{}a_{2},$ $\cdots,a_{k},~{}b_{2},~{}a_{i}a_{j},~{}1\leq
i,~{}j\leq k\\}\subseteq A$ , non-zero positive element $b_{1}$ of $A$, there
exist a projection $p\in A$ and a ${\rm C^{*}}$-subalgebra $B$ of $A$ with
$1_{B}=p$ and $B\in{\rm AT}\Omega$, such that

$(1)^{\prime}$ $\|b_{2}p-pb_{2}\|<\delta$, $\|a_{i}p-pa_{i}\|<\delta$,
$\|a_{i}a_{j}p-pa_{i}a_{j}\|<\delta$, for all $1\leq i,~{}j\leq k$,

$(2)^{\prime}$ $pb_{2}p\in_{\delta}B$, $pa_{i}p\in_{\delta}B$,
$pa_{i}a_{j}p\in_{\delta}B$, for all $1\leq i,~{}j\leq k$,

$(3)^{\prime}$ $1-p\precsim b_{1}$, and

$(4)^{\prime}$ $\|pa_{i}p\|\geq\|a_{i}\|-\varepsilon$,
$\|pa_{i}a_{j}p\|\geq\|a_{i}a_{j}\|-\varepsilon$, for all $1\leq i,~{}j\leq
k$.

By $(2)^{\prime}$ there exist $a_{i}^{\prime},~{}b_{2}^{\prime}\in B$ such
that

$\|pa_{i}p-a_{i}^{\prime}\|<\delta,\|b_{2}-b_{2}^{\prime}\|<\delta$

for all $1\leq i\leq k$. Since $A$ is a amenable $\rm{C^{*}}$-algebra and
$B\subseteq A$, by Theorem 2.15, there exists a completely positive
contractive linear map $\alpha^{\prime}:A\to B$ such that

$\|\alpha^{\prime}(a_{i}^{\prime})-a_{i}^{\prime}\|<\delta,~{}\|\alpha^{\prime}(a_{i}^{\prime}a_{j}^{\prime})-a_{i}^{\prime}a_{j}\|<\delta$

for all $1\leq i,~{}j\leq k$.

Since $B\in\rm{AT}\Omega$, by Theorem 2.12, for
$\mathcal{H^{\prime}}=\\{a_{1}^{\prime},a_{2}^{\prime},\cdots,a_{k}^{\prime}\\}$,
any $\delta^{\prime\prime}>0$ (with $\delta^{\prime\prime}<\delta$), positive
element $(b_{2}^{\prime}-\delta)_{+}$ (we can assume that
$(b_{2}^{\prime}-\delta)_{+}\neq 0$), there exist a ${\rm C^{*}}$-algebra $D$
in $\Omega$ and completely positive contractive linear maps
$\alpha^{\prime\prime}:B\to D$ and completely positive contractive linear maps
$\beta_{n}^{\prime\prime}:D\to B$, and $\gamma_{n}^{\prime\prime}:B\to
B\cap\beta_{n}^{\prime\prime}(D)^{\perp}$ such that

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

$(2)^{\prime\prime}$
$\|a_{i}^{\prime}-\gamma_{n}^{\prime\prime}(a_{i}^{\prime})-\beta_{n}^{\prime\prime}(\alpha^{\prime\prime}(a_{i}^{\prime}))\|<\delta^{\prime\prime}$,
for all $1\leq i\leq k$, and for all $n\in{\mathbb{N}}$,

$(3)^{\prime\prime}$ $\alpha^{\prime\prime}$ is an
$\mathcal{H^{\prime}}$-$\delta^{\prime\prime}$ approximate embedding,

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

$(5)^{\prime\prime}$
$\gamma_{n}^{\prime\prime}(1_{B})\precsim(b_{2}^{\prime}-\delta)_{+}$, for all
$n\in\mathbb{N}$.

Define $\gamma_{n}:A\to A$, by
$\gamma_{n}(a)=(1-p)a(1-p)+\gamma_{n}^{\prime\prime}\alpha^{\prime}(pap)$,
$\alpha:A\to D$, by $\alpha(a)=\alpha^{\prime\prime}\alpha^{\prime}(pap)$ and
$\beta_{n}:D\to A$ by $\beta_{n}(a)=\beta_{n}^{\prime\prime}(a)$. Then we have
$\gamma_{n}$, $\beta_{n}$ and $\alpha$ are completely positive contractive
linear maps (since for any $a\in A$,
$(1-p)a(1-p)\gamma_{n}^{\prime\prime}\alpha^{\prime}(pap)=0$).

We also have

$\begin{array}[]{ll}&$(1)$~{}~{}~{}~{}~{}\|a_{i}-\gamma_{n}(a_{i})-\beta_{n}\alpha(a_{i})\|\\\
&=\|a_{i}-(1-p)a_{i}(1-p)-\gamma^{\prime\prime}\alpha^{\prime}(pa_{i}p)-\beta_{n}^{\prime\prime}\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}p)\|\\\
&\leq\|a_{i}-(1-p)a_{i}(1-p)-pa_{i}p\|+\|pa_{i}p-\gamma^{\prime\prime}\alpha^{\prime}(pa_{i}p)-\beta_{n}^{\prime\prime}\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}p)\|\\\
&\leq
2\delta+\|pa_{i}p-a_{i}^{\prime}\|+\|a_{i}^{\prime}-\gamma^{\prime\prime}\alpha^{\prime}(pa_{i}p)-\beta_{n}^{\prime\prime}\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}p)\|\\\
&=2\delta+\delta+\|a_{i}^{\prime}-\gamma^{\prime\prime}(a_{i}^{\prime})-\beta_{n}^{\prime\prime}\alpha^{\prime\prime}(a_{i}^{\prime})\|+\|\gamma^{\prime\prime}\alpha^{\prime}(pa_{i}p)-\gamma^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime})\|\\\
&+\|\gamma^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime})-\gamma^{\prime\prime}(a_{i}^{\prime})\|+\|\beta_{n}^{\prime\prime}\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}p)-\beta_{n}^{\prime\prime}\alpha^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime})\|\\\
&+\|\beta_{n}^{\prime\prime}\alpha^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime})-\beta_{n}^{\prime\prime}\alpha^{\prime\prime}(a_{i}^{\prime})\|<3\delta+\delta^{\prime\prime}+\delta+\delta+\delta+\delta<\varepsilon,\end{array}$

for all $1\leq i,j\leq k$.

$\begin{array}[]{ll}&$(2)$~{}~{}~{}~{}~{}\|\alpha(a_{i}a_{j})-\alpha(a_{i})\alpha(a_{j})\|\\\
&=\|\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}a_{j}p)-\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}p)\alpha^{\prime\prime}\alpha^{\prime}(pa_{j}p)\|\\\
&\leq\|\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}a_{j}p)-\alpha^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime}a_{j}^{\prime})\|+\|\alpha^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime}a_{j}^{\prime})-\alpha^{\prime\prime}(\alpha^{\prime}(a_{i}^{\prime})\alpha^{\prime}(a_{j}^{\prime}))\|\\\
&+\|\alpha^{\prime\prime}(\alpha^{\prime}(a_{i}^{\prime})\alpha^{\prime}(a_{j}^{\prime}))-\alpha^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime})\alpha^{\prime\prime}\alpha^{\prime}(a_{j}^{\prime})\|\\\
&+\|\alpha^{\prime\prime}\alpha^{\prime}(a_{i}^{\prime})\alpha^{\prime\prime}\alpha^{\prime}(a_{j}^{\prime})-\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}p)\gamma^{\prime\prime}\alpha^{\prime}(pa_{j}p)\|\\\
&<\delta+\delta+2\delta+\delta<\varepsilon,\end{array}$

for all $1\leq i,j\leq k$.

We also have
$\|\alpha(a_{i})\|=\|\alpha^{\prime\prime}\alpha^{\prime}(pa_{i}p)\|\geq\|\alpha^{\prime}(pa_{i}p)\|-\delta^{\prime\prime}\geq\|\alpha^{\prime}(a_{i})\|-\delta^{\prime\prime}-\delta\geq\|a_{i}^{\prime}\|-\delta^{\prime\prime}-2\delta\geq\|pa_{i}p\|-\delta^{\prime\prime}-3\delta\geq\|a_{i}\|-\delta^{\prime\prime}-4\delta>\|a_{i}\|-\varepsilon.$

Therefore, $\alpha$ is a $\mathcal{G}$-$\varepsilon$ approximate embedding.

$(3)$ $\lim_{n\to\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$ and
$\lim_{n\to\infty}\|\beta_{n}(x)\|=\|x\|$, for all $x,~{}y\in D$, and

$(4)$
$\gamma_{n}(1_{A})=1-p+\gamma_{n}^{\prime\prime}\alpha^{\prime}(p1_{A}p)\precsim
b_{1}\oplus\gamma_{n}^{\prime\prime}(1_{B})\precsim
b_{1}\oplus(b_{2}^{\prime}-\delta)_{+}\precsim b_{1}+b_{2}\precsim b$, for all
$n\in\mathbb{N}$. ∎

######  Theorem 3.2.

Let $\Omega$ be a class of unital $\rm C^{*}$-algebras which have tracial
nuclear dimension at most $n$. Let $A$ be an infinite dimensional simple
unital $\rm C^{*}$-algebra and $A$ is asymptotical tracially in $\Omega$ (i.e.
$A\in\rm AT\Omega$). Then ${\rm Tdim_{nuc}}(A)\leq n$.

###### Proof.

By Theorem 2.8, we must show that for any finite subset $G\subseteq A$, for
any $\varepsilon>0$, and for any nonzero positive element $a\in A_{+}$, there
exist a finite dimensional $\rm C^{*}$-algebra $F$, a completely positive
contractive map $\alpha:A\rightarrow F$, a nonzero piecewise contractive
$n$-decomposable completely positive map $\beta:F\rightarrow A$, and a
completely positive contractive map $\gamma:A\rightarrow
A\cap\beta^{\perp}(F)$ such that

$(1)$ $\|x-\gamma(x)-\beta\alpha(x)\|<\varepsilon$, for all $x\in G$, and

$(2)$ $(\gamma(1_{A})-\varepsilon)_{+}\precsim a$.

Since $A$ is an infinite dimensional simple unital $\rm C^{*}$-algebra, there
exist positive elements $a_{1},~{}a_{2}\in A_{+}$ of norm one such that
$a_{1}a_{2}=0$, $a_{1}\sim a_{2}$ and $a_{1}+a_{2}\precsim a$.

Give $\varepsilon^{\prime}>0$, with
$G^{\prime}=G\cup\\{a_{1},~{}a_{2},~{}1_{A}\\}\subseteq A$, since $A$ is
asymptotical tracially in $\Omega$ (i.e., $A\in\rm AT\Omega$), there exist a
$\rm C^{*}$-algebra $B$ in ${\Omega}$, completely positive contractive maps
$\overline{\alpha}:A\rightarrow B$, $\overline{\beta_{n}}:B\rightarrow A$, and
$\overline{\gamma_{n}}:A\rightarrow A\cap\beta_{n}^{\perp}(B)(n\in\mathbb{N})$
such that

$(1^{\prime})$
$\|x-\overline{\gamma_{n}}(x)-\overline{\beta_{n}}\overline{\alpha}(x)\|<\varepsilon^{\prime}$,
for all $x\in G^{\prime}$, and for all $n\in\mathbb{N}$,

$(2^{\prime})$ $\overline{\alpha}$ is a
$(G^{\prime},~{}\varepsilon^{\prime})$-approximate embedding,

$(3^{\prime})$
$\mathop{\lim}\limits_{n\rightarrow\infty}\|\overline{\beta_{n}}(xy)-\overline{\beta_{n}}(x)\overline{\beta_{n}}(y)\|=0$
and
$\mathop{\lim}\limits_{n\rightarrow\infty}\|\overline{\beta_{n}}(x)\|=\|x\|$
for all $x,~{}y\in B$, and

$(4^{\prime})$ $\overline{\gamma_{n}}(1_{A})\precsim a_{1}\sim a_{2}$, for all
$n\in\mathbb{N}$.

Since $a_{2}\in A_{+}$ and $\overline{\alpha}$ is a
$(G^{\prime},\varepsilon^{\prime})$-approximate embedding, then
$\overline{\alpha}(a_{2})\in B_{+}$ and $\overline{\alpha}(a_{2})\neq 0$. We
may assume that $(\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+}\neq 0$.

Since $B\in{\Omega}$, ${\rm Tdim_{nuc}}(B)\leq n$, with
$\\{\overline{\alpha}(x),x\in G^{\prime}\\}\subseteq B$, with
$\varepsilon^{\prime}$, and nonzero positive element
$(\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+}\in B_{+}$, there exist a
finite dimensional $\rm C^{*}$-algebra $F$, a completely positive contractive
map $\alpha^{\prime}:B\rightarrow F$, a nonzero piecewise contractive
$n$-decomposable completely positive map $\beta^{\prime}:F\rightarrow B$, and
a completely positive contractive map $\gamma^{\prime}:B\rightarrow
B\cap\beta^{\perp}(F)$ such that

$(1^{\prime\prime})$
$\|\overline{\alpha}(x)-\gamma^{\prime}\overline{\alpha}(x)-\beta^{\prime}\alpha^{\prime}\overline{\alpha}(x)\|<\varepsilon^{\prime}$,
for all $\\{\overline{\alpha}(x),x\in G^{\prime}\\}$, and

$(2^{\prime\prime})$
$(\gamma^{\prime}(1_{B})-\varepsilon)_{+}\precsim(\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+}$.

Define $\alpha:A\rightarrow F$, by
$\alpha(x)=\alpha^{\prime}\overline{\alpha}(x)$, $\varphi:F\rightarrow A$, by
$\varphi(x)=\overline{\beta_{n}}\beta^{\prime}(x)$.

For $x,~{}y\in F_{i}$ with $xy=0$, then
$\varphi(x)\varphi(y)=\overline{\beta_{n}}\beta^{\prime}(x)\overline{\beta_{n}}\beta^{\prime}(y)$.
By $(3^{\prime})$, for $\delta>0$, there exists $N>0$, such that $n>N$, we
have
$\|\overline{\beta_{n}}\beta^{\prime}(x)\overline{\beta_{n}}\beta^{\prime}(y)-\overline{\beta_{n}}(\beta^{\prime}(x)\beta^{\prime}(y))\|<\delta$.
Since $\beta^{\prime}|_{F_{i}}$ is completely positive contractive order zero
map, then $\varphi|_{F_{i}}$ is $\delta$-almost order zero map. By Theorem
2.19, there exists $\beta:F\rightarrow A$ satisfying $\beta|_{F_{i}}$ is a
completely positive contractive order zero map such that
$\|\beta-\varphi\|\leq\varepsilon$, then $\beta$ is a nonzero piecewise
contractive $n$-decomposable completely positive map.

Define $\gamma:A\rightarrow A\cap\beta^{\perp}(F)$, by
$\gamma(x)=\overline{\gamma_{n}}(x)+\overline{\beta_{n}}\gamma^{\prime}\overline{\alpha}(x)$.
Since $\overline{\gamma_{n}}(A)\perp\overline{\beta_{n}}(B)$, we have $\gamma$
is a completely positive contractive.

Then for all $x\in G$, we have

$\|x-\gamma(x)-\beta\alpha(x)\|=\|x-\overline{\gamma_{n}}(x)-\overline{\beta_{n}}\gamma^{\prime}\overline{\alpha}(x)-\beta\alpha(x)\|$

$\leq\|x-\overline{\gamma_{n}}(x)-\overline{\beta_{n}}\gamma^{\prime}\overline{\alpha}(x)-\varphi\alpha(x)\|+\|\varphi\alpha(x)-\beta\alpha(x)\|$

$=\|x-\overline{\gamma_{n}}(x)-\overline{\beta_{n}}\gamma^{\prime}\overline{\alpha}(x)-\overline{\beta_{n}}\beta^{\prime}\alpha^{\prime}\overline{\alpha}(x)\|+\|\varphi\alpha(x)-\beta\alpha(x)\|$

$\leq\|x-\overline{\gamma_{n}}(x)-\overline{\beta_{n}}\overline{\alpha}(x)\|+\|\overline{\beta_{n}}\overline{\alpha}(x)-\overline{\beta_{n}}\gamma^{\prime}\overline{\alpha}(x)-\overline{\beta_{n}}\beta^{\prime}\alpha^{\prime}\overline{\alpha}(x)\|$

$+\|\varphi\alpha(x)-\beta\alpha(x)\|$

$\leq\varepsilon^{\prime}+\varepsilon^{\prime}+\varepsilon^{\prime}$

$\leq\varepsilon$.

Since
$(\gamma^{\prime}(1_{B})-\varepsilon)_{+}\precsim(\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+}$,
for sufficiently small $\delta^{\prime}>0$, there exist $v\in B$ such that

$\|v(\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+}v^{*}-(\gamma^{\prime}(1_{B})-\varepsilon)_{+}\|<\delta^{\prime},$

then, we have

$\|\overline{\beta_{n}}(v(\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+}v^{*})-\overline{\beta_{n}}((\gamma^{\prime}(1_{B})-\varepsilon)_{+})\|<\delta^{\prime}.$

By $(3^{\prime})$, with sufficiently larger $n$, we have

$\|\overline{\beta_{n}}(v)\overline{\beta_{n}}((\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+})\overline{\beta_{n}}(v^{*})-\beta_{n}((\gamma^{\prime}(1_{B})-\varepsilon)_{+})\|<\delta^{\prime}.$

Therefore, we have

$\overline{\beta_{n}}((\gamma^{\prime}(1_{B})-2\varepsilon)_{+})\precsim\overline{\beta_{n}}((\gamma^{\prime}(1_{B})-3\delta^{\prime}-\varepsilon)_{+})\precsim\overline{\beta_{n}}((\overline{\alpha}(a_{2})-\varepsilon^{\prime})_{+}.$

By $(1^{\prime})$,
$\|a_{2}-\overline{\gamma_{n}}(a_{2})-\overline{\beta_{n}}\overline{\alpha}(a_{2})\|<\varepsilon^{\prime}$,
then we have

$(\overline{\beta_{n}}\overline{\alpha}(a_{2})-\varepsilon^{\prime})+\overline{\gamma_{n}}(a_{2})\precsim
a_{2},$

so we have

$(\overline{\beta_{n}}\overline{\alpha}(a_{2})-\varepsilon^{\prime})+\precsim
a_{2}.$

We also have

$(\gamma(1_{A})-3\varepsilon)_{+}\precsim\overline{\gamma_{n}}(1_{A})+(\overline{\beta_{n}}\gamma^{\prime}\overline{\alpha}(1_{A})-2\varepsilon)_{+}$

$\precsim\overline{\gamma_{n}}(1_{A})+(\overline{\beta_{n}}\gamma^{\prime}(1_{B})-2\varepsilon)_{+}$

$\precsim
a_{1}\oplus\overline{\beta_{n}}(\overline{\alpha}(a_{2})-\varepsilon)_{+}$

$\precsim a_{1}+a_{2}\precsim a$.

Since $\alpha=\alpha^{\prime}\overline{\alpha}$, $\alpha^{\prime}$ and
$\overline{\alpha}$ are completely positive contractive maps, then $\alpha$ is
a completely positive contractive map.

Therefore, ${\rm Tdim_{nuc}}(A)\leq n$.

∎

######  Theorem 3.3.

([16].) Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras such that
$\Omega$ is closed under passing to unital hereditary ${\rm C^{*}}$-subalgebra
and tensoring matrix algebras. Let $A\in$ $\rm{TA}\Omega$ be an infinite-
dimensional simple unital ${\rm C^{*}}$-algebra. Suppose that
$\alpha:G\to{\rm{Aut}}(A)$ is an action of a finite group $G$ on $A$ which has
tracial Rokhlin property. Then the crossed product ${\rm C^{*}}$-algebra ${\rm
C^{*}}(G,A,\alpha)$ belongs to $\rm{TA}\Omega$.

The following Corollary also can be obtained by a result of Fu and Lin of [18]
and a result of Elliott, Fan and Fang of [6] and Theorem 3.3).

######  Corollary 3.4.

Let $A$ be an infinite dimensional simple separable amenable unital ${\rm
C^{*}}$-algebra with ${\rm Tdim_{nuc}}(A)\leq n$. Suppose that
$\alpha:G\to{\rm Aut}(A)$ is an action of a finite group $G$ on $A$ which has
the tracial Rokhlin property. Then ${\rm Tdim_{nuc}}({{\rm
C^{*}}(G,A,\alpha)})\leq n$.

###### Proof.

By Theorem 3.1, Theorem 3.2 and Theorem 3.3.

∎

######  Theorem 3.5.

Let $\Omega$ be a class of unital $\rm C^{*}$-algebras which have the second
type tracial nuclear dimension at most $n$. Let $A$ be an infinite dimensional
simple unital $\rm C^{*}$-algebra and $A$ is asymptotical tracially in
$\Omega$ (i.e. $A\in\rm AT\Omega$). Then ${\rm T^{2}dim_{nuc}}(A)\leq n$.

###### Proof.

We must show that for any positive subset $G\subseteq A_{+}$ (we may assume
that $G\subseteq A_{+}^{1}$), for any $\varepsilon>0$, and for any nonzero
positive element $a\in A_{+}$, there exist a finite dimension $\rm
C^{*}$-algebra $F=F_{0}\oplus\cdots\oplus F_{n}$ and completely positive maps
$\psi:A\rightarrow F$ and $\varphi:F\rightarrow A$ such that

$(1)$ for all $x\in G$, exists $x^{\prime}\in A_{+}$, such that
$x^{\prime}\precsim a$, and $\|x-x^{\prime}-\varphi\psi(x)\|<\varepsilon$,

$(2)$ $\|\psi\|\leq 1$, and

$(3)$ $\varphi|_{F_{i}}$ is a completely positive contractive order zero map,
for $i=1,~{}\cdots,~{}n$.

Since $A$ is an infinite dimensional simple unital $\rm C^{*}$-algebra, there
exist positive elements $a_{1},~{}a_{2}\in A_{+}$ of norm one such that
$a_{1}a_{2}=0$, $a_{1}\sim a_{2}$ and $a_{1}+a_{2}\precsim a$.

Give $\varepsilon^{\prime}>0$, with
$G^{\prime}=G\cup\\{a_{1},~{}a_{2}\\}\subseteq A$, since $A$ is asymptotical
tracially in $\Omega$ (i.e. $A\in\rm AT\Omega$), there exist a $\rm
C^{*}$-algebra $B$ in $\Omega$, and completely positive contractive maps
$\alpha:A\rightarrow B$, $\beta_{n}:B\rightarrow A$, and
$\gamma_{n}:A\rightarrow A\cap\beta_{n}^{\perp}(B)(n\in\mathbb{N})$ such that

$(1^{\prime})$ $\|x-\gamma_{n}(x)-\beta_{n}\alpha(x)\|<\varepsilon^{\prime}$,
for all $x\in G^{\prime}$, and for all $n\in{\mathbb{N}}$,

$(2^{\prime})$ $\alpha$ is a $(G^{\prime},\varepsilon^{\prime})$-approximate
embedding,

$(3^{\prime})$
$\mathop{\lim}\limits_{n\rightarrow\infty}\|\beta_{n}(xy)-\beta_{n}(x)\beta_{n}(y)\|=0$
and $\mathop{\lim}\limits_{n\rightarrow\infty}\|\beta_{n}(x)\|=\|x\|$ for all
$x,~{}y\in B$, and

$(4^{\prime})$ $\gamma_{n}(1_{A})\precsim a_{1}\sim a_{2}$, for all
$n\in{\mathbb{N}}$.

Since $\alpha$ is a $(G^{\prime},\varepsilon^{\prime})$-approximate embedding,
then $\alpha(a_{2})$ is nonzero. We may assume that
$(\alpha(a_{2})-\varepsilon^{\prime})_{+}\neq 0$.

Since $B\in\Omega$, then ${\rm T^{2}dim_{nuc}}(B)\leq n$, with
$\\{\alpha(x),x\in G^{\prime}\\}\subseteq B$, with $\varepsilon^{\prime}$, and
nonzero positive element $\alpha(a_{2})\in B_{+}$, there exist a finite
dimension $\rm C^{*}$-algebra $F=F_{0}\oplus\cdots\oplus F_{n}$ and completely
positive maps $\psi^{\prime}:B\rightarrow F$, and
$\varphi^{\prime}:F\rightarrow B$ such that

$(1^{\prime\prime})$ for all $x\in G^{\prime}$, there exists
$\alpha(x)^{\prime}\in B_{+}$, such that
$\alpha(x)^{\prime}\precsim\alpha(a_{2})$ and and
$\|\alpha(x)-\alpha(x)^{\prime}-\varphi^{\prime}\psi^{\prime}\alpha(x)\|<\varepsilon^{\prime}$,

$(2^{\prime\prime})$ $\|\psi^{\prime}\|\leq 1$, and

$(3^{\prime\prime})$ $\varphi^{\prime}|_{F_{i}}$ is a completely positive
contractive order zero map, for $i=1,~{}\cdots,~{}n$.

Define $\psi:A\rightarrow F$, by $\psi(a)=\psi^{\prime}\alpha(a)$,
$\overline{\varphi}:F\rightarrow A$, by
$\overline{\varphi}(a)=\beta_{n}\varphi^{\prime}(a)$ and
$x^{\prime}=\gamma_{n}(x)+(\beta_{n}\alpha(x)^{\prime}-2\varepsilon)_{+}$,
since $\gamma_{n}(x),\beta_{n}\alpha(x)^{\prime}\in A_{+}$, then
$x^{\prime}\in A_{+}$.

For $x,~{}y\in F_{i}$ with $xy=0$, then
$\overline{\varphi}(x)\overline{\varphi}(y)=\beta_{n}\varphi^{\prime}(x)\beta_{n}\varphi^{\prime}(y).$
By $(3^{\prime})$, for $\delta>0$, there exists sufficiently large $N>0$, such
that $n>N$, one has
$\|\beta_{n}\varphi^{\prime}(x)\beta_{n}\varphi^{\prime}(y)-\beta_{n}(\varphi^{\prime}(x)\varphi^{\prime}(y))\|<\delta$.
Since $\varphi^{\prime}|_{F_{i}}$ is completely positive contractive order
zero map, then $\overline{\varphi}|_{F_{i}}$ is $\delta$-almost order zero
map. By Theorem 2.19, there exists $\varphi:F\rightarrow A$ satisfying
$\varphi|_{F_{i}}$ is a completely positive contractive order zero map and
$\|\varphi-\overline{\varphi}\|\leq\varepsilon$.

Then for all $x\in G$, we have

$\|x-x^{\prime}-\varphi\psi(x)\|=\|x-\gamma_{n}(x)-(\beta_{n}\alpha(x)^{\prime}-2\varepsilon)_{+}-\varphi\psi(x)$

$\leq
2\varepsilon+\|x-\gamma_{n}(x)-\beta_{n}\alpha(x)^{\prime}-\overline{\varphi}\psi(x)\|+\|\overline{\varphi}\psi(x)-\varphi\psi(x)\|$

$\leq
3\varepsilon+\|x-\gamma_{n}(x)-\beta_{n}\varphi^{\prime}\psi^{\prime}\alpha(x)-\beta_{n}\alpha(x)^{\prime}\|$

$\leq
3\varepsilon+\|x-\gamma_{n}(x)-\beta_{n}\alpha(x)\|+\|\beta_{n}\alpha(x)-\beta_{n}\varphi^{\prime}\psi^{\prime}\alpha(x)-\beta_{n}\alpha(x)^{\prime}\|$

$\leq
3\varepsilon+\varepsilon^{\prime}+\|\alpha(x)-\varphi^{\prime}\psi^{\prime}\alpha(x)-\alpha(x)^{\prime}\|$

$\leq 3\varepsilon+\varepsilon^{\prime}+\varepsilon^{\prime}$

$\leq 5\varepsilon$.

Since $\alpha(x)^{\prime}\precsim\alpha(a_{2})$, then there exists $v\in A$
such that

$\|v\alpha(a_{2})v^{*}-\alpha(x)^{\prime}\|<\varepsilon^{\prime}.$

By $(3^{\prime})$, we have

$\|\beta_{n}(v\alpha(a_{2})v^{*})-\beta_{n}\alpha(x)^{\prime}\|<\varepsilon^{\prime},$

by $(3^{\prime})$,

$\|\beta_{n}(v)\beta_{n}\alpha(a_{2})\beta_{n}(v^{*})-\beta_{n}\alpha(x)^{\prime}\|<2\varepsilon^{\prime}.$

Therefore, we have

$(\beta_{n}\alpha(x)^{\prime}-2\varepsilon)_{+}\precsim(\beta_{n}\alpha(a_{2})-\varepsilon^{\prime})_{+}.$

Since
$\|a_{2}-\gamma_{n}(a_{2})-\beta_{n}\alpha(a_{2})\|<\varepsilon^{\prime}$, we
have

$(\beta_{n}\alpha(a_{2})-\varepsilon^{\prime})_{+}+\gamma_{n}(a_{2})\precsim
a_{2}.$

Then, we have

$(\beta_{n}\alpha(a_{2})-\varepsilon^{\prime})_{+}\precsim a_{2}.$

Therefore, we have

$x^{\prime}=\gamma_{n}(x)+(\beta_{n}\alpha(x)^{\prime}-2\varepsilon)_{+}$

$\precsim\gamma_{n}(1_{A})\oplus(\beta_{n}\alpha(a_{2})-\varepsilon^{\prime})_{+}\precsim\gamma_{n}(1_{A})\oplus
a_{2}$

$\precsim a_{1}+a_{2}\precsim a$.

Since $\psi=\psi^{\prime}\alpha$ and $\|\psi^{\prime}\|\leq 1$, then $\alpha$
is a completely positive contractive map.

Therefore, ${\rm T^{2}dim_{nuc}}(A)\leq n$. ∎

######  Corollary 3.6.

Let $A$ be an infinite-dimensional simple separable amenable unital ${\rm
C^{*}}$-algebra with ${\rm T^{2}dim_{nuc}}(A)\leq n$. Suppose that
$\alpha:G\to{\rm Aut}(A)$ is an action of a finite group $G$ on $A$ which has
the tracial Rokhlin property. Then ${\rm T^{2}dim_{nuc}}{{\rm
C^{*}}(G,A,\alpha)}\leq n$.

###### Proof.

By Theorem 3.1, Theorem 3.2 and Theorem 3.5.

∎

## References

  * [1] R. Antoine, F. Perera and H. Thiel, Tensor products and regularity properties of Cuntz semigroups. Mem. Amer. Math. Soc., 251 (2018), no. 1199; 199 pages.
  * [2] R. Antoine, F. Perera, L. Robert and H. Thiel,${C^{*}}$-algebras of stable rank one and their Cuntz semigroups. Duke Math. J., 171 (2022), 33–99.
  * [3] P. Ara, F. Perera, and A. Toms, $K$-theory for operator algebras. Classification of ${C^{*}}$-algebras, Aspects of operator algebras and applications, 1–71, Contemp. Math., 534, American Mathematical Society, Providence, RI, 2011.
  * [4] A. Connes, Outer conjugacy class of automorphisms of factors, Ann. Sci. Ecole Norm. Sup., 8 (1975), 383–420.
  * [5] G. A. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensional algebras, J. Algebra, 38 (1976), 29–44.
  * [6] G. A. Elliott, Q. Fan, and X. Fang, Certain properties of tracial approximation ${C^{*}}$-algebras, C. R. Math. Rep. Acad. Sci. Canada, 40, (2018), 104–133.
  * [7] G. A. Elliott and G. Gong, On the classification of ${C^{*}}$-algebras of real rank zero, II, Ann. Math., 144 (1996), 497–610.
  * [8] G. A. Elliott, G. Gong, and L. Li, On the classification of simple inductive limit ${C^{*}}$-algebras II: The isomorphism theorem, Invent. Math., 168 (2007), 249–320.
  * [9] G. A. Elliott, G. Gong, H. Lin, and Z. Niu, Simple stably projectionless ${C^{*}}$-algebras with generalized tracial rank one, J. Noncommut. Geom. 14, (2020), 251–347.
  * [10] G. A. Elliott and A. Toms, Regularity properties in the classification program for separable amenable ${C^{*}}$-algebras, Bull. Amer. Math. Soc., 45 (2008), 229–245.
  * [11] G. A. Elliott, G. Gong, H. Lin, and Z. Niu, On the classification of simple amenable ${C^{*}}$-algebras with finite decomposition rank, II, arXiv: 1507.03437.
  * [12] G. A. Elliott and Z. Niu, On tracial approximation, J. Funct. Anal., 254 (2008), 396–440.
  * [13] Q. Fan, Some ${C^{*}}$-algebra properties preserved by tracial approximation, Israel J. Math., 195 (2013), 545–563.
  * [14] Q. Fan and X. Fang, Non-simple tracial approximation, Houston J. of Math. 37 (2011), 1249–1263.
  * [15] Q. Fan and X. Fang, Stable rank one and real rank zero for crossed products by finite group actions with the tracial Rokhlin property, Chinese Ann. Math. Ser. B, 30 (2009), 179–186.
  * [16] Q. Fan and X. Fang, Crossed products by finite group actions with a certain tracial Rokhlin property, Acta Math. Scientia, 38 (2018), 829–842.
  * [17] X. Fu, Tracial nuclear dimension of ${C^{*}}$-algebras, Ph.D. thesis, East China Normal University, 2018.
  * [18] X. Fu and H. Lin, Tracial approximation in simple $C^{*}$-algebras, Canad. J. Math., First View, 26 February 2021, pp. 1–63.
  * [19] G. Gong, On the classification of simple inductive limit ${C^{*}}$-algebras. I. The reduction theorem, Doc. Math., 7 (2002), 255–461.
  * [20] G. Gong and H. Lin, On classification of non-unital simple amenable ${C^{*}}$-algebras, II, Journal of Geometry and Physics, 158 , Article 103865, 102 pages, (2020).
  * [21] G. Gong and H. Lin, On classification of non-unital simple amenable ${C^{*}}$-algebras, III, arXiv: 2010. 01788.
  * [22] G. Gong, H. Lin, and Z. Niu, Classification of finite simple amenable $\mathcal{Z}$-stable ${C^{*}}$-algebras, I: ${C^{*}}$-algebra with generalized tracial rank one, C. R. Math. Rep. Acad. Sci. Canada, 42 (2020), 63–450.
  * [23] G. Gong, H. Lin, and Z. Niu, Classification of finite simple amenable $\mathcal{Z}$-stable ${C^{*}}$-algebras, II: ${C^{*}}$-algebras with rational generalized tracial rank one, C. R. Math. Rep. Acad. Sci. Canada, 42 (2020), 451–539.
  * [24] H. H. Lee and H. Osaka Tracially sequentially-split *-homomorphisms between ${C^{*}}$-algebras, arXiv:1707. 07377.
  * [25] R. Herman and A. Ocneanu, Stability for integer actions on UHF ${C^{*}}$-algebras, J. Funct. Anal., 59 (1984), 132–144.
  * [26] I. Hirshberg and J. Orovitz, Tracially $\mathcal{Z}$-absorbing ${C^{*}}$-algebras, J. Funct. Anal., 265 (2013), 765–785.
  * [27] S. Hu, L. Hin, and Y. Xue, The tracial topological rank of $C^{*}$-algebras (II), Indiana Univ. Math. J. 53 (2004), 1579-1606.
  * [28] S. Hu, L. Hin, and Y. Xue, Tracial limit of $C^{*}$-algebras, Acta Mathematica Sinica, English Series 19 (2003), 535¨C556.
  * [29] M. Izumi, Finite group actions on ${C^{*}}$-algebras with Rohlin property. I, Duke Math. J., 122 (2004), 233–280.
  * [30] E. Kirchberg and M. Rørdam, Central sequence ${C^{*}}$-algebras and tensorial absorption of the Jiang-Su algebra, J. Reine Angew. Math., 695 (2014), 175–214.
  * [31] A. Kishimoto, The Rohlin property for shifts on UHF algebras, J. Reine Angew. Math., 465 (1995), 183–196.
  * [32] H. Lin, Tracially AF ${C^{*}}$-algelbras, Trans. Amer. Math. Soc., 353 (2001), 683–722.
  * [33] H. Lin, The tracial topological rank of ${C^{*}}$-algebras, Proc. London Math. Soc., 83 (2001), 199–234.
  * [34] H. Lin, An introduction to the classification of amenable ${C^{*}}$-algebras, World Scientific, New Jersey, London, Singapore, Hong Kong, 2001.
  * [35] H. Osaka and N. C. Phillips, Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property, Ergodic Theory Dynam. Systems, 26 (2006), 1579–1621.
  * [36] H. Osaka and N. C. Phillips, Crossed products by finite group actions with the Rokhlin property, Math. Z., 270 (2012), 19–42.
  * [37] E. Ortega, M. Rørdam, and H. Thiel, The Cuntz semigroup and comparison of open projections, J. Funct. Anal., 260 (2011), 3474–3493.
  * [38] C. Peligrad and L. Zsido, Open projections of ${C^{*}}$-algebras: comparison and regularity, Operator Theoretical Methods, Timisoara, 1998, Theta Foundation Bucharest, 2000.
  * [39] N. C. Phillips, The tracial Rokhlin property for actions of finite groups on ${C^{*}}$-algebras, Amer. J. Math., 133 (2011), 581–636.
  * [40] N. C. Phillips, Large subalgebras, arXiv: 1408. 5546v2.
  * [41] M. Rørdam, Classification of certain infinite simple ${C^{*}}$-algebras, J. Funct. Anal., 131 (1995), 415–458.
  * [42] M. Rørdam and W. Winter, The Jiang-Su algebra revisited, J. Reine Angew. Math., 642 (2010), 129–155.
  * [43] H. Thiel, Ranks of operators in simple ${C^{*}}$-algebras with stable rank one. Commun. Math. Phys., 377 (2020), 37–76.
  * [44] A. Tikuisis, S. White, and W. Winter, Quasidiagonality of nuclear ${C^{*}}$-algebras, Ann. Math., 185 (2017), 229–284.
  * [45] W. Winter and J. Zacharias, The nuclear dimension of $C^{*}$-algebra, Adv. math., 224 (2010), 461–498.