# certain properties of generalized tracially approximated
${\rm\textbf{C}^{*}}$-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.

We show that the following properties of unital ${\rm C^{*}}$-algebra in a
class of $\Omega$ are preserved by unital simple ${\rm C^{*}}$-algebra in the
class of $\rm WTA\Omega$: $(1)$ uniform property $\Gamma$, $(2)$ a certain
type of tracial nuclear dimension at most $n$, $(3)$ weakly $(m,n)$-divisible.

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

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 [9]. A major next step was the classification of simple AH
algebras without dimension growth (in the real rank zero case see [12], and in
the general case see [13]). This led eventually to the classification of
simple separable amenable ${\rm C^{*}}$-algebras with finite nuclear dimension
in the UCT class (see [30], [36], [15], [27], [28], [41], [14], [25], and
[26]).

A crucial intermediate step was Lin’s axiomatization of Elliott-Gong’s
decomposition theorem for simple AH algebras of real rank zero (classified by
Elliott-Gong in [12]) and Gong’s decomposition theorem ([24]) for simple AH
algebras (classified by Elliott-Gong-Li in [13]). For this purpose, Lin
introduced the concepts of TAF and TAI ([32] and [33]). Instead of assuming
inductive limit structure, Lin started with a certain abstract (tracial)
approximation property. Elliott and Niu in [16] considered this notion of
tracial approximation by other classes of unital ${\rm C^{*}}$-algebras than
the finite-dimensional ones for TAF and the interval algebras for TAI.

Large and centrally large subalgebras were introduced in [37] and [5] by
Phillips and Archey as abstractions of Putnam’s orbit breaking subalgebra of
the crossed product algebra ${\rm C}^{*}(X,\mathbb{Z},\sigma)$ of the Cantor
set by a minimal homeomorphism in [38].

Inspired by centrally large subalgebras and tracial approximation ${\rm
C^{*}}$-algebras, Elliott, Fan and Fang introduced a class of unital weakly
tracial approximation ${\rm C^{*}}$-algebras in [11]. The notion generalizes
both Archey and Phillips’s centrally large subalgebras and Lin’s notion of
tracial approximation.

Let $\Omega$ be a class of ${\rm C^{*}}$-algebras. Elliott, Fan, and Fang in
[11] introduced the class of ${\rm C^{*}}$-algebras which can be weakly
tracially approximated by ${\rm C^{*}}$-algebras in $\Omega$, and denote this
class by ${\rm WTA}\Omega$ (see Definition 2.4).

Let $\Omega$ be a class of finite dimension ${\rm C^{*}}$-algebras. Let $A$ be
an infinite-dimensional unital simple $\rm C^{*}$-algebra and let $B\subseteq
A$ be a centrally large subalgebra of $A$ such that $B$ has tracial
topological rank zero. In [18], Fan, Fang and Wang show that $A\in{\rm
WTA}\Omega$.

In [11], Elliott, Fan, and Fang show that the following properties of unital
${\rm C^{*}}$-algebras in a class $\Omega$ are inherited by unital simple
${\rm C^{*}}$-algebras in the class of ${\rm WTA}\Omega$:

$(1)$ tracially $\mathcal{Z}$-absorbing,

$(2)$ tracial nuclear dimension at most $n$,

$(3)$ the property $\rm SP$, and

$(4)$ $m$-almost divisible.

In this paper, we shall prove the following results:

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras which have uniform
property $\Gamma$ (see Definition 2.7). Then $A$ has uniform property $\Gamma$
for any simple unital stably finite ${\rm C^{*}}$-algebra $A\in{\rm
WTA}\Omega$ (Theorem 3.1). Let $\Omega$ be a class of unital nuclear ${\rm
C^{*}}$-algebras which have a type of tracial nuclear dimension at most $n$
(see Definition 2.6). Then $A$ has the type of tracial nuclear dimension at
most $n$ for any simple unital ${\rm C^{*}}$-algebra $A\in{\rm WTA}\Omega$
(Theorem 3.4). Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras which
are weakly $(m,n)$-divisible (see Definition 2.2). Let $A\in{\rm WTA}\Omega$
be a simple unital stably finite $C^{*}$-algebra such that for any integer
$n\in\mathbb{N}$ the ${\rm C^{*}}$-algebra ${\rm M}_{n}(A)$ belongs to the
class ${\rm WTA}\Omega$. Then $A$ is secondly weakly $(m,n)$-divisible (see
Definition 2.3) (Theorem 3.7).

As applications, the following known results follow from these results.

Let $A$ be an infinite-dimensional stably finite unital simple separable ${\rm
C^{*}}$-algebra. Let $B\subseteq A$ be a centrally large subalgebra in $A$
such that $B$ has uniform property $\Gamma$. Then $A$ has uniform property
$\Gamma$. This result was obtained by Fan and Zhang in [19]

Let $\Omega$ be a class of stably finite unital ${\rm C^{*}}$-algebras such
that for any $B\in\Omega$, $B$ has uniform property $\Gamma$. Then $A$ has
uniform property $\Gamma$ for any simple unital ${\rm C^{*}}$-algebra
$A\in\rm{TA}\Omega$. This result was obtained by Fan and Zhang in [19].

Let $A$ be a unital simple stably finite separable ${\rm C^{*}}$-algebra. Let
$B\subseteq A$ be a centrally large subalgebra of $A$ such that $B$ is weakly
$(m,n)$-divisible. Then $A$ is secondly weakly $(m,n)$-divisible. This result
was obtained by Fan, Fang, and Zhao in [17].

##  2\. Preliminaries and definitions

Let $A$ be a ${\rm C^{*}}$-algebra, and let ${\rm M}_{n}(A)$ denote the ${\rm
C^{*}}$-algebra of $n\times n$ matrices with entries elements of $A$. Let
${\rm M}_{\infty}(A)$ denote the algebraic inductive limit of the sequence
$({\rm M}_{n}(A),\phi_{n}),$ where $\phi_{n}:{\rm M}_{n}(A)\to{\rm
M}_{n+1}(A)$ is the canonical embedding as the upper left-hand corner block.
Let ${\rm M}_{\infty}(A)_{+}$ (respectively, ${\rm M}_{n}(A)_{+}$) denote the
positive elements of ${\rm M}_{\infty}(A)$ (respectively, ${\rm M}_{n}(A)$).
Given $a,b\in{\rm M}_{\infty}(A)_{+},$ one says that $a$ is Cuntz
subequivalent to $b$ (written $a\precsim b$) if there is a sequence
$(v_{n})_{n=1}^{\infty}$ of elements of ${\rm M}_{\infty}(A)$ such that

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

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

The object ${\rm Cu}(A):={(A\otimes{{K}}})_{+}/\sim$ will be called the Cuntz
semigroup of $A$. (See [8].) Observe that any $a,b\in{\rm M}_{\infty}(A)_{+}$
are Cuntz equivalent to orthogonal elements $a^{\prime},b^{\prime}\in{\rm
M}_{\infty}(A)_{+}$ (i.e., $a^{\prime}b^{\prime}=0$), and so ${\rm Cu}(A)$
becomes an ordered semigroup when equipped with the addition operation

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

whenever $ab=0$, 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 either $a$ 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 $f(t)={\max(0,t-\varepsilon)}$, $t\in\sigma(a)$.

The following facts are well known.

######  Theorem 2.1.

([3], [29], [37], [40].) Let $A$ be a ${\rm C^{*}}$-algebra.

$(1)$ Let $a,b\in A_{+}$ and any $\varepsilon>0$ be such that
$\|a-b\|<\varepsilon$. Then $(a-\varepsilon)_{+}\precsim b$.

$(2)$ Let $a,p$ be positive elements in $A$ with $p$ a projection. If
$p\precsim a,$ and $p$ is not equivalent to $a$, then there is a nonzero
element $b$ in $A$ such that $bp=0$ and $b+p\precsim a$.

$(3)$ Let $a$ be a positive element of ${\rm C^{*}}$-algebra $A$ and $a$ is
not Cuntz equivalent to a projection. Let $\delta>0$, and let $g\in
C_{0}(0,1]$ be a non-negative function with $g=0$ on $(\delta,1),$ $g>0$ on
$(0,\delta)$, and $\|g\|=1$. Then $g(a)\neq 0$ and
$(a-\delta)_{+}+g(a)\precsim a.$

The property of weakly $(m,n)$-divisible was introduced by Kirchberg and
Rørdam in [31].

######  Definition 2.2.

([31].) Let $A$ be a unital ${\rm C^{*}}$-algebra. Let $m,n\geq 1$ be two
integers. $A$ is said to be weakly $(m,n)$-divisible, if for every $a\in{\rm
M}_{\infty}(A)_{+}$ and any $\varepsilon>0$, there exist elements
$x_{1},x_{2},\cdots,x_{n}\in{\rm M}_{\infty}(A)_{+}$, such that $m\left\langle
x_{j}\right\rangle\leq\left\langle a\right\rangle$, for all $j=1,2,\cdots,n$
and $\left\langle(a-\varepsilon)_{+}\right\rangle\leq\left\langle
x_{1}\right\rangle+\left\langle x_{2}\right\rangle+\cdots+\left\langle
x_{n}\right\rangle$.

######  Definition 2.3.

Let $A$ be a unital ${\rm C^{*}}$-algebra. Let $m,n\geq 1$ be two integers.
$A$ is said to be secondly weakly $(m,n)$-divisible, if for every $a\in{\rm
M}_{\infty}(A)_{+}$ and any $\varepsilon>0$, there exist elements
$x_{1},x_{2},\cdots,x_{n}\in{\rm M}_{\infty}(A)_{+}$ such that $m\left\langle
x_{j}\right\rangle\leq\left\langle a\right\rangle+\left\langle a\right\rangle$
for all $j=1,2,\cdots,n$, and
$\left\langle(a-\varepsilon)_{+}\right\rangle\leq\left\langle
x_{1}\right\rangle+\left\langle x_{2}\right\rangle+\cdots+\left\langle
x_{n}\right\rangle$.

Let $\Omega$ be a class of unital ${\rm C^{*}}$-algebras. Elliott, Fan, and
Fang defined as follows the class of ${\rm C^{*}}$-algebras which can be
weakly tracially approximated by ${\rm C^{*}}$-algebras in $\Omega$, and
denoted this class by ${\rm WTA}\Omega$ in [11].

######  Definition 2.4.

([11].) A simple unital ${\rm C^{*}}$-algebra $A$ is said to belong to the
class ${\rm WTA}\Omega$ if, for any $\varepsilon>0$, any finite subset
$F\subseteq A$, and any non-zero element $a\geq 0$, there exist a projection
$p\in A$, an element $g\in A$ with $0\leq g\leq 1$, and a unital ${\rm
C^{*}}$-subalgebra $B$ of $A$ with $g\in B,~{}1_{B}=p$, and $B\in\Omega$, such
that

$(1)$ $(p-g)x\in_{\varepsilon}B,~{}x(p-g)\in_{\varepsilon}B$ for all $x\in F$,

$(2)$ $\|(p-g)x-x(p-g)\|<\varepsilon$ for all $x\in F$,

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

$(4)$ $\|(p-g)a(p-g)\|\geq\|a\|-\varepsilon$.

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

It follows from the definitions and by the proof of Theorem 4.1 of [16] that
if $A$ is a simple unital ${\rm C^{*}}$-algebra and $A\in{\rm TA}\Omega$, then
$A\in{\rm WTA}\Omega$. Furthermore, if $\Omega=\\{B\\}$, and $B\subseteq A$ is
a centrally large subalgebra of $A$, then $A\in{\rm WTA}\Omega$.

Winter and Zacharias introduced the notion of nuclear dimension for ${\rm
C^{*}}$-algebras in [42].

######  Definition 2.5.

([42].) A ${\rm C^{*}}$-algebra $A$ has nuclear dimension at most $n$, and
denoted by ${\rm dim_{nuc}}\leq n$, if there exists a net
$(F_{\lambda},\psi_{\lambda},\varphi_{\lambda})_{\lambda\in\Lambda}$ such that
the $F_{\lambda}$ are finite-dimensional ${\rm C^{*}}$-algebras, and such that
$\psi_{\lambda}:A\to F_{\lambda}$ and $\varphi_{\lambda}:F_{\lambda}\to A$ are
completely positive maps satisfying

$(1)$ $\psi_{\lambda}\varphi_{\lambda}(a)\to a$ uniformly on finite subsets of
$A$,

$(2)$ $\|\psi_{\lambda}\|\leq 1$,

$(3)$ for each $\lambda$, $F_{\lambda}$ decomposes into $n+1$ ideals
$F_{\lambda}={F_{\lambda}}^{0}+\cdots+{F_{\lambda}}^{n}$ such that
$\varphi_{\lambda}|_{{F_{\lambda}}^{i}}$ is completely positive contractive
order zero maps (which means preserves orthogonality, i.e.,
$\varphi_{\lambda}(e)\varphi_{\lambda}(f)=0$ for all
$e,~{}f\in{{F_{\lambda}}^{i}}$ with $ef=0$) for $i=0,~{}\cdots,~{}n$.

Inspired by Hirshberg and Orovitz’s tracial $\mathcal{Z}$-absorption in [29],
Fu introduced some notion of tracial nuclear dimension in his doctoral
dissertation [22] (see also [23]).

######  Definition 2.6.

([22].) Let $A$ be a $\rm C^{*}$-algebra. Let $n\in\mathbb{N}$ be an integer.
$A$ is said to have second type of 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 contractive completely positive order zero map
for $i=1,\cdots,n$.

Inspired by Fu’s second type of tracial nuclear dimension in [22], we shall
introduce a new type of tracial nuclear dimension for unital $\rm
C^{*}$-algebra.

######  Definition 2.7.

Let $A$ be a unital $\rm C^{*}$-algebra. Let $n\in\mathbb{N}$ be an integer.
$A$ will said to have a new type of tracial nuclear dimension at most $n$, 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
$\|x-x^{\prime}-\varphi\psi(x)\|<\varepsilon$,

$(2)$ $(1_{A}-\varphi\psi(1_{A})-\varepsilon)_{+}\precsim a$,

$(3)$ $\|\psi\|\leq 1$, and

$(4)$ $\varphi|_{F_{i}}$ is a contractive completely positive order zero map
for $i=1,\cdots,n$.

Let $A$ be a unital $\rm C^{*}$-algebra. It is easy to know that
${\rm{T^{2}dim_{nuc}}}(A)\leq n$ implies $A$ has the new type of tracial
nuclear dimension at most $n$.

Uniform property $\Gamma$ was introduced by J. Castillejos, S. Evington, A.
Tikuisis, S. White, and W. Winter, which was used to prove that
$\mathcal{Z}$-stable imply that finite nuclear dimension in [6].

Examples of separable nuclear ${\rm C^{*}}$-algebra with uniform property
$\Gamma$ are by now abundant. Kerr and Szabó establish uniform property
$\Gamma$ for crossed product ${\rm C^{*}}$-algebras arising from a free action
with the small boundary property of an infinite amenable group on a compact
metrisable space see Theorem 9.4 in [20].

We recall the equivalent local refinement of Property $\Gamma$ from
Proposition 2.4 of [7].

######  Theorem 2.8.

([7].) Let $A$ be a separable ${\rm C^{*}}$-algebra with $T(A)$ nonempty and
compact. Then the following are equivalent:

$(1)$ $A$ has uniform property $\Gamma$.

$(2)$ For any finite subset $F\subseteq A$, any $\varepsilon>0$, and any
integer $n\in\mathbb{N}$, there exist pairwise orthogonal positive
contractions $e_{1},\cdots,e_{n}\in A$ such that for $i=1,\cdots,n$, and $a\in
F$, we have $\|e_{i}a-ae_{i}\|<\varepsilon$ and

$\sup_{\tau\in T(A)}\|\tau(ae_{i})-\frac{1}{n}\tau(a)\|<\varepsilon.$

##  3\. The main results

######  Theorem 3.1.

Let $\Omega$ be a class of unital separable $\rm C^{*}$-algebra such that
$T(B)$ is nonempty and compact and $B$ has uniform property $\Gamma$ for any
$B\in\Omega$. Then $A$ has uniform property $\Gamma$ for any simple infinite-
dimensional separable unital stably finite $\rm C^{*}$-algebra $A\in{\rm
WTA}\Omega$.

###### Proof.

Since $A$ is a stably finite $\rm C^{*}$-algebra, then $T(A)$ is nonempty.
This together with the unitality of $A$ implies that $T(A)$ is compact. By
Theorem 2.8: (2), we need to show that for fixed finite subset
$F=\\{a_{1},a_{2},\cdots,a_{k}\\}$ of $A$ (we may assume that $\|a_{j}\|\leq
1$ for all $j=1,\cdots,k$), any $\varepsilon>0$, and any integer
$n\in\mathbb{N}$, there exist pairwise orthogonal positive contractions
$e_{1},\cdots,e_{n}\in A$ such that $\|e_{i}a_{j}-a_{j}e_{i}\|<\varepsilon$
and

$\sup_{\tau\in T(A)}\|\tau(a_{j}e_{i})-\frac{1}{n}\tau(a_{j})\|<\varepsilon,$

for $i=1,\cdots,n,$ and $j=1,\cdots,k$.

For $\varepsilon>0$, we choose $0<\delta<\varepsilon$ with
$X=[0,1],f(t)=t^{1/2},g(t)=(1-t)^{1/2}$ according to Lemma 2.5.11 in [34].

Since $A$ is an infinite-dimensional unital simple separable ${\rm
C^{*}}$-algebra, by Corollary 2.5 in [37], there exists a nonzero positive
element $a\in A$ such that $\delta>d_{\tau}(a)=\lim_{n\to\infty}\tau(a^{1/n})$
for any $\tau\in T(A)$.

For $F=\\{a_{1},a_{2},\cdots,a_{k}\\}$, any $\delta>0$, and nonzero $a\in
A_{+}$, since $A\in{\rm WTA}\Omega$, there exist a projection $p\in A$, an
element $g\in A$ with $\|g\|\leq 1$ and a unital $\rm C^{*}$-subalgebra $B$ of
$A$ with $g\in B,1_{B}=p$ and $B\in\Omega$ such that

$(1)$ $(p-g)x\in_{\delta}B,~{}x(p-g)\in_{\delta}B$ for any $x\in F$,

$(2)$ $\|(p-g)x-x(p-g)\|<\delta$ for any $x\in F$, and

$(3)$ $1_{A}-(p-g)\precsim a$.

By $(2)$, we have

$(1)^{\prime}$ $\|(1_{A}-(p-g))a_{j}-a_{j}(1_{A}-(p-g))\|<\delta$ for any
$j=1,\cdots,k$.

By $(3)$ and by Proposition 1.19 of [37], we have

$d_{\tau}(1_{A}-(p-g))\leq d_{\tau}(a)$

for any $\tau\in T(A)$. Since $d_{\tau}(a)<\delta$, and $\tau(1_{A}-(p-g))\leq
d_{\tau}(1_{A}-(p-g))$, we have

$\tau(1_{A}-(p-g))<\delta$

for any $\tau\in T(A)$.

By the choice of $\delta$, by $(2)$, $(1)^{\prime}$ and by Lemma 2.5.11 in
[34], one has

$\|(1_{A}-(p-g))^{1/2}a_{j}-a_{j}(1_{A}-(p-g))^{1/2}\|<\varepsilon,$ and

$\|(p-g)^{1/2}a_{j}-a_{j}(p-g)^{1/2}\|<\varepsilon.$

By $(1)$, there exists $a_{j}^{\prime}\in B$ such that

$(2)^{\prime}$ $\|(p-g)a_{j}-a_{j}^{\prime}\|<\delta.$

By $(1)^{\prime}$ and $(2)^{\prime}$, one has

$\|a_{j}-a_{j}^{\prime}-(1_{A}-(p-g))^{1/2}a_{j}(1_{A}-(p-g))^{1/2}\|$

$=\|(1_{A}-(p-g))a_{j}+(p-g)a_{j}-a_{j}^{\prime}-(1_{A}-(p-g))^{1/2}a_{j}(1_{A}-(p-g))^{1/2}\|$

$\leq\|(1_{A}-(p-g))a_{j}-(1_{A}-(p-g))^{1/2}a_{j}(1_{A}-(p-g))^{1/2}\|+\|(p-g)a_{j}-a_{j}^{\prime}\|$$<\varepsilon+\delta<2\varepsilon,$
and

$\|(p-g)^{1/2}a_{j}(p-g)^{1/2}-a_{j}^{\prime}\|\leq\|(p-g)^{1/2}a_{j}(p-g)^{1/2}-(p-g)a_{j}\|$$+\|(p-g)a_{j}-a_{j}^{\prime}\|$
$<\varepsilon+\delta<2\varepsilon$ for all $1\leq j\leq k$.

For $\varepsilon>0$, and any integer $n\in\mathbb{N}$, we choose
$\delta^{\prime\prime}=\delta^{\prime\prime}(\varepsilon,n)$ (with
$\delta^{\prime\prime}<\varepsilon$) sufficiently small such that satisfying
Lemma 2.5.12 in [34].

For $\delta^{\prime\prime}/2>0$, finite subset
$\\{a_{1}^{\prime},a_{2}^{\prime},\cdots,a_{k}^{\prime},(p-g),(p-g)^{1/2}\\}$
of $B$ and $n\in\mathbb{N}$, since $B$ has uniform property $\Gamma$, there
exist pairwise orthogonal positive contractions
$e_{1}^{\prime},\cdots,e_{n}^{\prime}\in B$ such that for
$i=1,\cdots,n,j=1,\cdots,k$, one has

$\|e_{i}^{\prime}a_{j}^{\prime}-a_{j}^{\prime}e_{i}^{\prime}\|<\delta^{\prime\prime}/2,\|e_{i}^{\prime}(p-g)-(p-g)e_{i}^{\prime}\|<\delta^{\prime\prime}/2,$
$\|e_{i}^{\prime}(p-g)^{1/2}-(p-g)^{1/2}e_{i}^{\prime}\|<\delta^{\prime\prime}/2,$

and

$\sup_{\tau\in
T(B)}\|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})\|<\delta^{\prime\prime}/2.$

Since
$\|(p-g)^{1/2}e_{i}^{\prime}-e_{i}^{\prime}(p-g)^{1/2}\|<\delta^{\prime\prime}$,
and $e_{i}^{\prime}e_{j}^{\prime}=0,i\neq j$, so we have

$\|(p-g)^{1/2}e_{i}^{\prime}(p-g)^{1/2}(p-g)^{1/2}e_{j}^{\prime}(p-g)^{1/2}\|$

$\leq\|(p-g)^{1/2}e_{i}^{\prime}(p-g)^{1/2}(p-g)^{1/2}e_{j}^{\prime}(p-g)^{1/2}-(p-g)e_{i}^{\prime}(p-g)^{1/2}e_{j}^{\prime}(p-g)^{1/2}\|$
$+\|(p-g)e_{i}^{\prime}(p-g)^{1/2}e_{j}^{\prime}(p-g)^{1/2}-(p-g)e_{i}^{\prime}e_{j}^{\prime}(p-g)\|$

$<\delta^{\prime\prime}/2+\delta^{\prime\prime}/2=\delta^{\prime\prime}$.

By the choice of $\delta$, since $(p-g)^{1/2}e_{i}^{\prime}(p-g)^{1/2}$ is a
contraction, by the proof the Lemma 2.5.12 in [34], one can fine pairwise
orthogonal positive contractions $e_{i}$ $(i=1,\cdots,n)$, such that

$\|(p-g)^{1/2}e_{i}^{\prime}(p-g)^{1/2}-e_{i}\|<\varepsilon.$

We have

$\|a_{j}e_{i}-a_{j}^{\prime}e_{i}^{\prime}\|$$\leq\|a_{j}e_{i}-a_{j}(p-g)^{1/2}e_{i}^{\prime}(p-g)^{1/2}\|$

$+\|a_{j}(p-g)^{1/2}e_{i}^{\prime}(p-g)^{1/2}-a_{j}(p-g)^{1/2}(p-g)^{1/2}e_{i}^{\prime}\|$

$+\|a_{j}(p-g)^{1/2}(p-g)^{1/2}e_{i}^{\prime}-(p-g)^{1/2}a_{j}(p-g)^{1/2}e_{i}^{\prime}\|$

$+\|(p-g)^{1/2}a_{j}(p-g)^{1/2}e_{i}^{\prime}-a_{j}^{\prime}e_{i}^{\prime}\|\leq
4\varepsilon$.

With the same argument, one has

$\|e_{i}a_{j}-e_{i}^{\prime}a_{j}^{\prime}\|<4\varepsilon$

for $i=1,\cdots,n,j=1,\cdots,k$.

Since
$\|e_{i}^{\prime}a_{j}^{\prime}-a_{j}^{\prime}e_{i}^{\prime}\|<\delta^{\prime\prime}/2$,
so one has

$\|a_{j}e_{i}-e_{i}a_{j}\|$

$\leq\|a_{j}e_{i}-a_{j}^{\prime}e_{i}^{\prime}\|+\|a_{j}^{\prime}e_{i}^{\prime}-e_{i}^{\prime}a_{j}^{\prime}\|+\|e_{i}^{\prime}a_{j}^{\prime}-e_{i}a_{j}\|$

$<4\varepsilon+4\varepsilon+\delta^{\prime\prime}/2<9\varepsilon,$ for
$i=1,\cdots,n,j=1,\cdots,k$.

Since $\|a_{j}e_{i}-a_{j}^{\prime}e_{i}^{\prime}\|<4\varepsilon$ for any
$\tau\in T(A)$, one has

$|\tau(a_{j}e_{i})-\tau(a_{j}^{\prime}e_{i}^{\prime})|<4\varepsilon.$

Since
$\|a_{i}-a_{i}^{\prime}-(1_{A}-(p-g))^{1/2}a_{j}(1_{A}-(p-g))^{1/2}\|<2\varepsilon,$
we have

$|\tau(a_{j})-\tau(a_{j}^{\prime})-\tau((1_{A}-(p-g))^{1/2}a_{j}(1_{A}-(p-g))^{1/2})|<2\varepsilon.$

Therefore, we have

$|\tau(a_{j}e_{i})-\frac{1}{n}\tau(a_{j})|$

$=|\tau(a_{j}e_{i})-\tau(a_{j}^{\prime}e_{i}^{\prime})|+|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})|+|(\frac{1}{n}\tau(a_{j}^{\prime})-\frac{1}{n}\tau(a_{j})|$

$\leq
4\varepsilon+|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})|+\frac{1}{n}(2\varepsilon+\tau((1-(p-g))^{1/2}a_{j}(1-(p-g))^{1/2}))$

$\leq
4\varepsilon+|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})|+\frac{1}{n}(2\varepsilon+\tau(1-(p-g))$

$\leq
5\varepsilon+|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})|$.

Therefore, one has

$\sup_{\tau\in T(A)}|\tau(a_{j}e_{i})-\frac{1}{n}\tau(a_{j})|$

$\leq\sup_{\tau\in
T(A)}|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})|+5\varepsilon$

$\leq\frac{1}{\tau(p)}\sup_{\tau\in
T(B)}|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})|+5\varepsilon$

$\leq\frac{1}{1-\delta}\sup_{\tau\in
T(B)}|\tau(a_{j}^{\prime}e_{i}^{\prime})-\frac{1}{n}\tau(a_{j}^{\prime})|+5\varepsilon$

$<5\varepsilon+\frac{\delta^{\prime\prime}}{1-\delta}$ $<6\varepsilon.$

By Theorem 2.11:(2), $A$ has uniform property $\Gamma$. ∎

The following two corollaries were obtained by Fan and Zhang in [19].

######  Corollary 3.2.

([19]) Let $A$ be an infinite-dimensional stably finite unital simple ${\rm
C^{*}}$-algebra. Let $B\subseteq A$ be a centrally large subalgebra of $A$
such that $B$ has uniform property $\Gamma$. Then $A$ has uniform property
$\Gamma$.

######  Corollary 3.3.

([19]) Let $\Omega$ be a class of stably finite unital ${\rm C^{*}}$-algebras
such that for any $B\in\Omega$, $B$ has uniform property $\Gamma$. Then $A$
has uniform property $\Gamma$ for any simple unital ${\rm C^{*}}$-algebra
$A\in\rm{TA}\Omega$.

######  Theorem 3.4.

Let $\Omega$ be a class of unital nuclear ${\rm C^{*}}$-algebras which have
the new type of tracial nuclear dimension at most $n$ (in the sense of
Definition 2.7). Then $A$ has the new type of tracial nuclear dimension at
most $n$ for any simple unital ${\rm C^{*}}$-algebra $A\in{\rm WTA}\Omega$.

###### Proof.

We must show that for any finite positive subset $\mathcal{F}=\\{a_{1},a_{2},$
$\cdots,a_{k}\\}$ $\subseteq A$, for any $\varepsilon>0$, and for any nonzero
positive element $b\in A$, 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 $\overline{x}\in A_{+}$, such that
$\|x-\overline{x}-\varphi\psi(x)\|<\varepsilon$,

$(2)$ $(1_{A}-\varphi\psi(1_{A})-\varepsilon)_{+}\precsim b$,

$(3)$ $\|\psi\|\leq 1$, and

$(4)$ $\varphi|_{F_{i}}$ is a completely positive contractive order zero map
for $i=1,\cdots,n$.

By Lemma 2.3 of [34], there exist positive elements $b_{1},b_{2}\in A$ of norm
one such that $b_{1}b_{2}=0,b_{1}\sim b_{2}$ and $b_{1}+b_{2}\precsim b$.

Given $\varepsilon^{\prime}>0$, for $H=\mathcal{F}\cup\\{b\\}$, since
$A\in{\rm WTA}\Omega$, there exist a projection $p\in A$, an element $g\in A$
with $\|g\|\leq 1$ and a unital $\rm C^{*}$-subalgebra $B$ of $A$ with $g\in
B,1_{B}=p$ and $B\in\Omega$ such that

$(1)^{\prime}$
$(p-g)x\in_{\varepsilon^{\prime}}B,x(p-g)\in_{\varepsilon^{\prime}}B$ for any
$x\in H$,

$(2)^{\prime}$ $\|(p-g)x-x(p-g)\|<\varepsilon^{\prime}$ for any $x\in H$,

$(3)^{\prime}$ $1_{A}-(p-g)\precsim b_{1}\sim b_{2}$, and

$(4)^{\prime}$ $\|(p-g)b(p-g)\|\geq 1-\varepsilon^{\prime}$.

By $(2)^{\prime}$ and Lemma 2.5.11 of [34], with sufficiently small
$\varepsilon^{\prime}$, we can get

$(5)^{\prime}$ $\|(p-g)^{\frac{1}{2}}x-x(p-g)^{\frac{1}{2}}\|<\varepsilon$ for
any $x\in H$, and

$(6)^{\prime}$
$\|(1_{A}-(p-g))^{\frac{1}{2}}x-x(1_{A}-(p-g))^{\frac{1}{2}}\|<\varepsilon$
for any $x\in H$.

By $(1)^{\prime}$ and $(5)^{\prime}$, with sufficiently small
$\varepsilon^{\prime}$, there exist positive elements
$a^{\prime}_{1},\cdots,a^{\prime}_{n}\in B$ and a positive element
$b_{2}^{\prime}\in B$ such that

$\|(p-g)^{\frac{1}{2}}a_{i}(p-g)^{\frac{1}{2}}-a^{\prime}_{i}\|<\varepsilon$
for $1\leq i\leq k$, and
$\|(p-g)^{\frac{1}{2}}b_{2}(p-g)^{\frac{1}{2}}-b_{2}^{\prime}\|<\varepsilon$.

Therefore, one has

$\|a_{i}-a^{\prime}_{i}-(1_{A}-(p-g))^{\frac{1}{2}}a_{i}(1_{A}-(p-g))^{\frac{1}{2}}\|$

$\leq\|a_{i}-(p-g)a_{i}-(1_{A}-(p-g))a_{i}\|+\|(p-g)a_{i}-a^{\prime}_{i}\|$

$+\|(p-g)^{\frac{1}{2}}a_{i}(p-g)^{\frac{1}{2}}-(p-g)a_{i}\|$

$+\|(1_{A}-(p-g))a_{i}-(1_{A}-(p-g))^{\frac{1}{2}}a_{i}(1_{A}-(p-g))^{\frac{1}{2}}\|$

$<\varepsilon+\varepsilon+\varepsilon+\varepsilon=4\varepsilon$ for $1\leq
i\leq k$.

Since
$\|(p-g)^{\frac{1}{2}}b_{2}(p-g)^{\frac{1}{2}}-b_{2}^{\prime}\|<\varepsilon$,
by (1) of Theorem 2.1, we have

$(b_{2}^{\prime}-3\varepsilon)_{+}\precsim((p-g)^{\frac{1}{2}}b_{2}(p-g)^{\frac{1}{2}}-2\varepsilon)_{+}.~{}~{}~{}\hskip
11.38092pt\textbf{(3.4.1)}$

By $(4)^{\prime}$, one has

$\|(p-g)^{\frac{1}{2}}b_{2}(p-g)^{\frac{1}{2}}\|\geq\|(p-g)b_{2}(p-g)\|\geq
1-\varepsilon.$

Therefore, we have
$\|(b_{2}^{\prime}-3\varepsilon)_{+}\|\geq\|(p-g)^{\frac{1}{2}}b_{2}(p-g)^{\frac{1}{2}}\|+4\varepsilon\geq
1-5\varepsilon$, then, with $0<\varepsilon^{\prime}<\varepsilon<\frac{1}{5}$,
$(b_{2}^{\prime}-3\varepsilon)_{+}\neq 0$.

Define a completely positive contractive map
$\varphi^{\prime\prime}:A\rightarrow A$ by
$\varphi^{\prime\prime}(a)=(1_{A}-(p-g))^{\frac{1}{2}}a(1_{A}-(p-g))^{\frac{1}{2}}$.
Since $B$ is a nuclear $\rm C^{*}$-algebra, by Theorem 2.3.13 of [34], there
exists a contractive completely positive map $\psi^{\prime\prime}:A\rightarrow
B$ such that $\|\psi^{\prime\prime}(p-g)-(p-g)\|<\varepsilon$, and
$\|\psi^{\prime\prime}(a^{\prime}_{i})-a^{\prime}_{i}\|<\varepsilon$ for all
$1\leq i\leq n$.

Since $B\in\Omega$, then $B$ has the new type of tracial nuclear dimension at
most $n$, there exist a finite dimension $\rm C^{*}$-algebra
$F=F_{0}\oplus\cdots\oplus F_{n}$ and completely positive maps
$\psi^{\prime}:B\rightarrow F$, $\varphi^{\prime}:F\rightarrow B$ such that

$(1)^{\prime\prime}$ for any $a_{i}^{\prime}$ ($1\leq i\leq k$), there exists
$\overline{\overline{a_{i}^{\prime}}}\in B_{+}$, such that
$\|a_{i}^{\prime}-\overline{\overline{a_{i}^{\prime}}}-\varphi^{\prime}\psi^{\prime}(a_{i}^{\prime})\|<\varepsilon$,
and for $g\in B_{+}$, there exists $\overline{\overline{g}}\in B$, such that
$\|g-\overline{\overline{g}}-\varphi^{\prime}\psi^{\prime}(g)\|<\varepsilon.~{}~{}~{}~{}~{}~{}\hskip
11.38092pt\textbf{(3.4.2)}$

$(2)^{\prime\prime}$
$(p-\varphi^{\prime}\psi^{\prime}(p)-\varepsilon)_{+}\precsim(b_{2}^{\prime}-3\varepsilon)_{+}$,

$(3)^{\prime\prime}$ $\|\psi^{\prime}\|\leq 1$, and

$(4)^{\prime\prime}$ $\varphi^{\prime}|_{F_{i}}$ is a completely positive
contractive order zero map for $i=1,\cdots,n$.

Define $\varphi:F\rightarrow A$ by $\varphi(a)=\varphi^{\prime}(a)$ and
$\psi:A\rightarrow F$ by
$\psi(a)=\psi^{\prime}\psi^{\prime\prime}((p-g)^{\frac{1}{2}}a(p-g)^{\frac{1}{2}})$
and
$\overline{a_{i}}=\varphi^{\prime\prime}(a_{i})+\overline{\overline{a_{i}^{\prime}}}\in
A_{+}$ for $1\leq i\leq k$.

Then, one has

$(1_{A}-\varphi\psi(1_{A})-4\varepsilon)_{+}$

$=(1_{A}-\varphi^{\prime}\psi^{\prime}\psi^{\prime\prime}(p-g)-4\varepsilon)_{+}$

$\precsim
1_{A}-(\varphi^{\prime}\psi^{\prime}(p)-2\varepsilon)_{+}+\varphi^{\prime}\psi^{\prime}(g)$

$\precsim(1_{A}-p)+(p-\varphi^{\prime}\psi^{\prime}(p)-\varepsilon)_{+}+\varphi^{\prime}\psi^{\prime}(g)$

$\precsim(1_{A}-p)+(p-\varphi^{\prime}\psi^{\prime}(p)-\varepsilon)_{+}+\varphi^{\prime}\psi^{\prime}(g)+(\overline{\overline{g}}-\varepsilon)_{+}$

$\precsim(1_{A}-p)+g+(p-\varphi^{\prime}\psi^{\prime}(p)-\varepsilon)_{+}$ (by
(3.4.2))

$\precsim b_{1}\oplus(b_{2}^{\prime}-3\varepsilon)_{+}$ (by $(1)^{\prime}$)

$\precsim
b_{1}\oplus((p-g)^{\frac{1}{2}}b_{2}(p-g)^{\frac{1}{2}}-2\varepsilon)_{+}$ (by
(3.4.1))

$\precsim b_{1}+b_{2}\precsim b$.

One also has

$\|a_{i}-\overline{a_{i}}-\varphi\psi(a_{i})\|=\|a_{i}-\varphi^{\prime\prime}(a_{i})-\overline{\overline{a_{i}^{\prime}}}-\varphi^{\prime}\psi^{\prime}\psi^{\prime\prime}((p-g)^{\frac{1}{2}}a_{i}(p-g)^{\frac{1}{2}})\|$

$\leq\|a_{i}-(1_{A}-(p-g))^{\frac{1}{2}}a_{i}(1_{A}-(p-g))^{\frac{1}{2}}-\overline{\overline{a_{i}^{\prime}}}-\varphi^{\prime}\psi^{\prime}\psi^{\prime\prime}((p-g)^{\frac{1}{2}}a_{i}(p-g)^{\frac{1}{2}})\|$

$\leq\|a_{i}-a^{\prime}_{i}-(1_{A}-(p-g))^{\frac{1}{2}}a_{i}(1_{A}-(p-g))^{\frac{1}{2}}\|+\|a^{\prime}_{i}-\overline{\overline{a_{i}^{\prime}}}-\varphi^{\prime}\psi^{\prime}\psi^{\prime\prime}((p-g)^{\frac{1}{2}}a_{i}(p-g)^{\frac{1}{2}})\|$
$\leq
3\varepsilon+\|a^{\prime}_{i}-\overline{\overline{a_{i}^{\prime}}}-\varphi^{\prime}\psi^{\prime}(a_{i}^{\prime})\|$

$+\|\varphi^{\prime}\psi^{\prime}(a_{i}^{\prime})-\varphi^{\prime}\psi^{\prime}\psi^{\prime\prime}(a_{i}^{\prime})\|+\|\varphi^{\prime}\psi^{\prime}\psi^{\prime\prime}(a_{i}^{\prime})-\varphi^{\prime}\psi^{\prime}\psi^{\prime\prime}((p-g)^{\frac{1}{2}}a_{i}(p-g)^{\frac{1}{2}})\|$
$<3\varepsilon+2\varepsilon+2\varepsilon+2\varepsilon=9\varepsilon$.

Since
$\varphi^{\prime\prime},\varphi^{\prime},\psi^{\prime},\psi^{\prime\prime}$
are completely positive contractive maps, then $\varphi$ and $\psi$ are
completely positive maps.

For $(3)^{\prime\prime}$, $\varphi^{\prime}|_{F_{i}}$ is a completely positive
contractive order zero map for $i=0,1,\cdots,n$ and
$\varphi(a)=\varphi^{\prime}(a)$, then $\varphi|_{F_{i}}$ is contractive
completely positive order zero map for $i=0,1,\cdots,n$.

For any $x\in A$,
$\|\psi(x)\|=\|\psi^{\prime}\psi^{\prime\prime}((p-g)^{\frac{1}{2}}x(p-g)^{\frac{1}{2}})\|\leq\|\psi^{\prime}\|\|\psi^{\prime\prime}\|\|x\|$,
then $\|\psi\|\leq\|\psi^{\prime\prime}\|\|\psi^{\prime}\|\leq 1$.

Therefore, $A$ has the new type of tracial nuclear dimension at most $n$. ∎

######  Corollary 3.5.

Let $A$ be a unital simple ${\rm C^{*}}$-algebra. Let $B\subseteq A$ be a
centrally large nuclear subalgebra of $A$ such that $B$ has the new type of
tracial nuclear dimension at most $n$. Then $A$ has the new type of tracial
nuclear dimension at most $n$.

######  Corollary 3.6.

Let $\Omega$ be a class of unital nuclear ${\rm C^{*}}$-algebras such that for
any $B\in\Omega$, $B$ has the new type of tracial nuclear dimension at most
$n$. Then $A$ has the new type of tracial nuclear dimension at most $n$ for
any simple unital ${\rm C^{*}}$-algebra $A\in\rm{TA}\Omega$.

######  Theorem 3.7.

Let $\Omega$ be a class of unital $\rm C^{*}$-algebras such that $B$ is weakly
$(m,n)$-divisible (with $n\neq m$) (see Definition 2.2) for any $B\in\Omega$.
Let $A\in{\rm WTA}\Omega$ be a simple unital stably finite $\rm C^{*}$-algebra
such that for any integer $n\in\mathbb{N}$ the $\rm C^{*}$-algebra ${\rm
M}_{n}(A)$ belongs to the class ${\rm WTA}\Omega$. Then $A$ is secondly weakly
$(m,n)$-divisible (with $n\neq m$) (see Definition 2.3).

###### Proof.

For given $a\in{\rm M}_{\infty}(A)_{+},~{}\varepsilon>0$, we may assume that
$a\in A_{+}$, and $\|a\|=1$, (we have replaced ${\rm M}_{n}(A)$ containing $a$
given initially by $A$), we must show that there are
$x_{1},x_{2},\cdots,x_{n}\in{\rm M}_{\infty}(A)_{+}$ such that $m\langle
x_{j}\rangle\leq\langle a\rangle+\langle a\rangle$, and
$\langle(a-\varepsilon)_{+}\rangle\leq\langle x_{1}\rangle+\langle
x_{2}\rangle+\cdots+\langle x_{n}\rangle$ for all $j=1,2,\cdots,n$.

For any $\delta_{1}>0$, since $A\in{\rm WTA}\Omega$, there exist a projection
$p\in A$, an element $g\in A$ with $0\leq g\leq 1$, and a $\rm
C^{*}$-subalgebra $B$ of $A$ with $g\in B$, $1_{B}=p$, and $B\in\Omega$ such
that

(1) $(p-g)a\in_{\delta_{1}}B$, and

(2) $\|(p-g)a-a(p-g)\|<\delta_{1}$.

By (2), with sufficiently small $\delta_{1}$, by Lemma 2.5.11 (1) of [34], we
have

(3) $\|(p-g)^{\frac{1}{2}}a-a(p-g)^{\frac{1}{2}}\|<\varepsilon/3$, and

(4) $\|(1-(p-g))^{\frac{1}{2}}a-a(1-(p-g))^{\frac{1}{2}}\|<\varepsilon/3$.

By (1) and (2), with sufficiently small $\delta_{1}$, there exists a positive
element $a^{{}^{\prime}}\in B$ such that

(5)
$\|(p-g)^{\frac{1}{2}}a(p-g)^{\frac{1}{2}}-a^{{}^{\prime}}\|<\varepsilon/3$.

By (3), (4) and (5),

$\|a-a^{{}^{\prime}}-(1-(p-g))^{\frac{1}{2}}a(1-(p-g))^{\frac{1}{2}}\|$

$\leq\|a-(p-g)a-(1-(p-g))a\|+\|(p-g)a-(p-g)^{\frac{1}{2}}a(p-g)^{\frac{1}{2}}\|$

$+\|(1-(p-g))a-(1-(p-g))^{\frac{1}{2}}a(1-(p-g))^{\frac{1}{2}}\|$

$+\|(p-g)^{\frac{1}{2}}a(p-g)^{\frac{1}{2}}-a^{{}^{\prime}}\|$

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

Since $B$ is weakly $(m,n)$-divisible, and
$(a^{{}^{\prime}}-2\varepsilon)_{+}\in B_{+}$, there exist
$x_{1}^{{}^{\prime}},x_{2}^{{}^{\prime}},$ $\cdots,x_{n}^{{}^{\prime}}\in B$,
such that $\langle x_{j}^{{}^{\prime}}\rangle+\langle
x_{j}^{{}^{\prime}}\rangle+\cdots+\langle
x_{j}^{{}^{\prime}}\rangle\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle$,
and $\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{{}^{\prime}}\rangle$, where $\langle x_{j}^{{}^{\prime}}\rangle$
repeats $m$ times.

Since $B$ is weakly $(m,n)$-divisible, and
$(a^{{}^{\prime}}-\varepsilon)_{+}\in B$, there exist
$y_{1}^{{}^{\prime}},y_{2}^{{}^{\prime}},$ $\cdots,y_{n}^{{}^{\prime}}\in B$,
such that $\langle y_{j}^{{}^{\prime}}\rangle+\langle
y_{j}^{{}^{\prime}}\rangle+\cdots+\langle
y_{j}^{{}^{\prime}}\rangle\leq\langle(a^{{}^{\prime}}-\varepsilon)_{+}\rangle$,
and $\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
y_{i}^{{}^{\prime}}\rangle$, where $\langle y_{j}^{{}^{\prime}}\rangle$
repeats $m$ times.

Write $a^{{}^{\prime\prime}}=(1-(p-g))^{\frac{1}{2}}a(1-(p-g))^{\frac{1}{2}}$.

We divide the proof into two cases.

Case (1) We assume that $(a^{{}^{\prime}}-2\varepsilon)_{+}$ is Cuntz
equivalent to a projection.

(1.1) We assume that $(a^{{}^{\prime}}-3\varepsilon)_{+}$ is Cuntz equivalent
to a projection.

(1.1.1) We assume that $(a^{{}^{\prime}}-2\varepsilon)_{+}$ is Cuntz
equivalent to $(a^{{}^{\prime}}-3\varepsilon)_{+}$.

(1.1.1.1) If
$x_{1}^{{}^{\prime}},x_{2}^{{}^{\prime}},\cdots,x_{n}^{{}^{\prime}}\in B$ are
all Cuntz equivalent to projections, and
$\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{{}^{\prime}}\rangle$, then, by Theorem 2.1 (2), there exist some
integer $j$ and a nonzero projection $d$ such that $\langle
x_{j}^{{}^{\prime}}+d\rangle+\langle
x_{j}^{{}^{\prime}}+d\rangle+\cdots+\langle
x_{j}^{{}^{\prime}}+d\rangle\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle$,
where $\langle x_{j}^{{}^{\prime}}+d\rangle$ repeats $m$ times, otherwise,
this contradicts the stably finiteness of $A$ (since $m\neq n$ and $\rm
C^{*}$-algebra $A$ is stably finite).

For any $\delta_{2}>0$, since $A\in{\rm WTA}\Omega$, there exist a projection
$p_{1}\in A$, an element $g_{1}\in A$ with $0\leq g_{1}\leq 1$, and a $\rm
C^{*}$-subalgebra $D_{1}$ of $A$ with $g_{1}\in D_{1}$, $1_{D{{}_{1}}}=p_{1}$,
and $D_{1}\in\Omega$ such that

($1^{{}^{\prime}}$) $(p_{1}-g_{1})a^{{}^{\prime\prime}}\in_{\delta_{2}}D_{1}$,

($2^{{}^{\prime}}$)
$\|(p_{1}-g_{1})a^{{}^{\prime\prime}}-a^{{}^{\prime\prime}}(p_{1}-g_{1})\|<\delta_{2}$,
and

($3^{{}^{\prime}}$) $1-(p_{1}-g_{1})\preceq d$.

By ($1^{{}^{\prime}}$) and ($2^{{}^{\prime}}$), with sufficiently small
$\delta_{2}$, as above, via the analogues for (4), (5) and (6) for
$a^{{}^{\prime\prime}},~{}p_{1},$ and $g_{1}$, there exist a positive element
$a^{{}^{\prime\prime\prime}}\in D_{1}$, such that

$\|(p_{1}-g_{1})^{\frac{1}{2}}a^{{}^{\prime\prime}}(p_{1}-g_{1})^{\frac{1}{2}}-a^{{}^{\prime\prime\prime}}\|<\varepsilon/3,$
and
$\|a^{{}^{\prime\prime}}-a^{{}^{\prime\prime\prime}}-(1-(p_{1}-g_{1}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{1}-g_{1}))^{\frac{1}{2}}\|<\varepsilon.$

Since $D_{1}$ is weakly $(m,n)$-divisible, and
$(a^{{}^{\prime\prime\prime}}-2\varepsilon)_{+}\in D_{1}$, there exist
positive elements
$x_{1}^{{}^{\prime\prime}},x_{2}^{{}^{\prime\prime}},\cdots,x_{n}^{{}^{\prime\prime}}\in
D_{1}$, such that

$\langle x_{j}^{{}^{\prime\prime}}\rangle+\langle
x_{j}^{{}^{\prime\prime}}\rangle+\cdots+\langle
x_{j}^{{}^{\prime\prime}}\rangle\leq\langle(a^{{}^{\prime\prime\prime}}-2\varepsilon)_{+}\rangle$,
and
$\langle(a^{{}^{\prime\prime\prime}}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{{}^{\prime\prime}}\rangle$, where $\langle
x_{j}^{{}^{\prime\prime}}\rangle$ repeats $m$ times.

Since $a^{{}^{\prime}}\leq a^{{}^{\prime}}+a^{{}^{\prime\prime}},$ then
$\langle(a^{{}^{\prime}}-\varepsilon)_{+}\rangle\leq\langle(a^{{}^{\prime}}+a^{{}^{\prime\prime}}-\varepsilon)_{+}\rangle$,
and $\|a-a^{{}^{\prime}}-a^{{}^{\prime\prime}}\|<\varepsilon$, so one has
$\langle(a^{{}^{\prime}}+a^{{}^{\prime\prime}}-\varepsilon)_{+}\rangle\leq\langle
a\rangle$.

Therefore, one has

$\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle\leq\langle(a^{{}^{\prime}}-\varepsilon)_{+}\rangle\leq\langle
a\rangle.$

Write
$x=(1-(p_{1}-g_{1}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{1}-g_{1}))^{\frac{1}{2}}$.

Since $a^{{}^{\prime\prime\prime}}\leq a^{{}^{\prime\prime\prime}}+x$, then
$\langle(a^{{}^{\prime\prime\prime}}-2\varepsilon)_{+}\rangle\leq\langle(a^{{}^{\prime\prime\prime}}+x-\varepsilon)_{+}\rangle$,
and $\|a^{{}^{\prime\prime}}-a^{{}^{\prime\prime\prime}}-x\|<\varepsilon$,
which implies that

$\langle(a^{{}^{\prime\prime\prime}}-2\varepsilon)_{+}\rangle\leq\langle
a^{{}^{\prime\prime}}\rangle\leq\langle a\rangle.$

Therefore, we have

$\langle(x_{j}^{{}^{\prime}}\oplus d)\oplus
x_{j}^{{}^{\prime\prime}}\rangle+\langle(x_{j}^{{}^{\prime}}\oplus d)\oplus
x_{j}^{{}^{\prime\prime}}\rangle+\cdots+\langle(x_{j}^{{}^{\prime}}\oplus
d)+x_{j}^{{}^{\prime\prime}}\rangle$

$\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle+\langle(a^{{}^{\prime\prime\prime}}-2\varepsilon)_{+}\rangle\leq\langle
a\rangle+\langle a\rangle$, where $\langle(x_{j}^{{}^{\prime}}\oplus d)\oplus
x_{j}^{{}^{\prime\prime}}\rangle$ repeats $m$ times, and

$\langle x_{i}^{{}^{\prime}}\oplus x_{i}^{{}^{\prime\prime}}\rangle+\langle
x_{i}^{{}^{\prime}}\oplus x_{i}^{{}^{\prime\prime}}\rangle+\cdots\langle
x_{i}^{{}^{\prime}}\oplus x_{i}^{{}^{\prime\prime}}\rangle$,

$\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle+\langle(a^{{}^{\prime\prime\prime}}-2\varepsilon)_{+}\rangle\leq\langle
a\rangle+\langle a\rangle$, where $\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{{}^{\prime\prime}}\rangle$ repeats $m$ times for $1\leq i\leq n$ and
$i\neq j$.

We also have

$\langle(a-10\varepsilon)_{+}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{{}^{\prime\prime\prime}}-3\varepsilon)_{+}\rangle+\langle(1-(p_{1}-g_{1}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{1}-g_{1}))^{\frac{1}{2}}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{{}^{\prime\prime\prime}}-3\varepsilon)_{+}\rangle+\langle(1-(p_{1}-g_{1}))\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{{}^{\prime\prime\prime}}-3\varepsilon)_{+}\rangle+\langle
d\rangle$

$\leq\sum_{i=1,i\neq j}^{n}\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{{}^{\prime\prime}}\rangle+\langle(x_{j}^{{}^{\prime}}\oplus d)\oplus
x_{j}^{{}^{\prime\prime}}\rangle.$

There are the desired inequalities, with$\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{{}^{\prime\prime}}\rangle+\langle(x_{j}^{{}^{\prime}}\oplus d)\oplus
x_{j}^{{}^{\prime\prime}}\rangle$ in place of $\langle x_{i}\rangle$, and
$10\varepsilon$ in place of $\varepsilon$.

(1.1.1.2) If
$x_{1}^{{}^{\prime}},x_{2}^{{}^{\prime}},\cdots,x_{k}^{{}^{\prime}}\in B$ are
projections, and
$\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle<\sum_{i=1}^{n}\langle
x_{i}^{{}^{\prime}}\rangle$, then, by Theorem 2.1 (2), there exists a nonzero
projection $e$, such that
$\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle
e\rangle\leq\sum_{i=1}^{n}\langle x_{i}^{{}^{\prime}}\rangle$.

As in the part (1.1.1.1), since $A\in{\rm WTA}\Omega$, there exist a
projection $p_{2}\in A$, an element $g_{2}\in A$ with $0\leq g_{2}\leq 1$, and
a $\rm C^{*}$-subalgebra $D_{2}$ of $A$ with $g_{2}\in D_{2}$,
$1_{D_{2}}=p_{2}$, and $D_{2}\in\Omega$, by $(1)^{\prime}$, there exists a
positive element $a^{4}\in D_{2}$, such that

$1-(p_{2}-g_{2})\preceq e,$
$\|(p_{2}-g_{2})^{\frac{1}{2}}a^{{}^{\prime\prime}}(p_{2}-g_{2})^{\frac{1}{2}}-a^{4}\|<\varepsilon/3,$
and

$\|a^{{}^{\prime\prime}}-a^{4}-(1-(p_{2}-g_{2}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{2}-g_{2}))^{\frac{1}{2}}\|<\varepsilon.$

Also as in the part (1.1.1.1), we have
$\langle(a^{4}-2\varepsilon)_{+}\rangle\leq\langle a\rangle.$

Since $D_{2}$ is weakly $(m,n)$-divisible, $(a^{4}-2\varepsilon)_{+}\in
D_{2}$, there exist $x_{1}^{4},x_{2}^{4},$ $\cdots,x_{n}^{4}\in D_{2}$, such
that

$\langle x_{j}^{4}\rangle+\langle x_{j}^{4}\rangle+\cdots+\langle
x_{j}^{4}\rangle\leq\langle(a^{4}-2\varepsilon)_{+}\rangle$, and

$\langle(a^{4}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{4}\rangle$, where $\langle x_{j}^{4}\rangle$ repeats $m$ times.

Therefore, we have

$\langle x_{j}^{{}^{\prime}}\oplus x_{j}^{4}\rangle+\langle
x_{j}^{{}^{\prime}}\oplus x_{j}^{4}\rangle+\cdots+\langle
x_{j}^{{}^{\prime}}\oplus x_{j}^{4}\rangle$

$\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle+\langle(a^{4}-2\varepsilon)_{+}\rangle\leq\langle
a\rangle+\langle a\rangle$, where $\langle x_{j}^{{}^{\prime}}\oplus
x_{j}^{4}\rangle$ repeats $m$ time for $1\leq i\leq n$.

We also have

$\langle(a-10\varepsilon)_{+}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{4}-3\varepsilon)_{+}\rangle+\langle(1-(p_{2}-g_{2}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{2}-g_{2}))^{\frac{1}{2}}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{4}-3\varepsilon)_{+}\rangle+\langle(1-(p_{2}-g_{2}))\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{4}-3\varepsilon)_{+}\rangle+\langle
e\rangle$

$\leq\sum_{i=1}^{n}\langle x_{i}^{{}^{\prime}}\oplus x_{i}^{4}\rangle.$

There are the desired inequalities, with $\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{4}\rangle$ in place of $\langle x_{i}\rangle$, and $10\varepsilon$ in
place of $\varepsilon$.

(1.1.1.3) we assume that there is a purely positive element, and we assume
$x_{1}^{{}^{\prime}}$ is a purely positive element. As
$\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{{}^{\prime}}\rangle$, for any $\varepsilon>0$, there exists $\delta>0$
such that
$\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle\leq\langle(x_{1}^{{}^{\prime}}-\delta)_{+}\rangle\oplus\sum_{i=1}^{n}\langle
x_{i}^{{}^{\prime}}\rangle$. Since $x_{1}^{{}^{\prime}}$ is a purely positive
element, by Theorem 2.1 (3), there exists a nonzero positive element $s$ such
that $\langle(x_{1}^{{}^{\prime}}-\delta)_{+}\rangle+\langle
s\rangle\leq\langle x_{1}^{{}^{\prime}}\rangle$.

As in the part (1.1.1.1), since $A\in{\rm WTA}\Omega$, by $(1)^{\prime}$,
there exists a projection $p_{3}\in A$, an element $g_{3}\in A$ with $0\leq
g_{3}\leq 1$, and a $\rm C^{*}$-subalgebra $D_{3}$ of $A$ with $g_{3}\in
D_{3}$, $1_{D_{3}}=p_{3}$, and $D_{3}\in\Omega$ there exist a positive element
$a^{5}\in D_{3}$, such that

$1-(p_{3}-g_{3})\preceq s,$
$\|(p_{3}-g_{3})^{\frac{1}{2}}a^{{}^{\prime\prime}}(p_{3}-g_{3})^{\frac{1}{2}}-a^{5}\|<\varepsilon/3,$
and

$\|a^{{}^{\prime\prime}}-a^{5}-(1-(p_{3}-g_{3}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{3}-g_{3}))^{\frac{1}{2}}\|<\varepsilon.$

Also as in the part (1.1.1.1), we have
$\langle(a^{5}-2\varepsilon)_{+}\rangle\leq\langle a\rangle.$

Since $D_{3}$ is weakly $(m,n)$-divisible, $(a^{5}-2\varepsilon)_{+}\in
D_{3}$, there exist $x_{1}^{5},x_{2}^{5},$ $\cdots,x_{n}^{5}\in D_{3}$, such
that

$\langle x_{j}^{5}\rangle+\langle x_{j}^{5}\rangle+\cdots+\langle
x_{j}^{5}\rangle\leq\langle(a^{5}-2\varepsilon)_{+}\rangle$, and

$\langle(a^{5}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{5}\rangle$, where $\langle x_{j}^{5}\rangle$ repeats $m$ times.

Therefore, we have

$\langle x_{j}^{{}^{\prime}}\oplus x_{j}^{5}\rangle+\langle
x_{j}^{{}^{\prime}}\oplus x_{j}^{5}\rangle+\cdots+\langle
x_{j}^{{}^{\prime}}\oplus x_{j}^{5}\rangle$,

$\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle+\langle(a^{5}-2\varepsilon)_{+}\rangle\leq\langle
a\rangle+\langle a\rangle$, where $\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{5}\rangle$ repeats $m$ times for $1\leq i\leq n$.

We also have

$\langle(a-10\varepsilon)_{+}\rangle$

$\leq\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle+\langle(a^{5}-3\varepsilon)_{+}\rangle+\langle(1-(p_{3}-g_{3}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{3}-g_{3}))^{\frac{1}{2}}\rangle$

$\leq\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle+\langle(a^{5}-3\varepsilon)_{+}\rangle+\langle(1-(p_{3}-g_{3}))\rangle$

$\leq\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle+\langle(a^{5}-3\varepsilon)_{+}\rangle+\langle
s\rangle$ $\leq\sum_{i=1}^{n}\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{5}\rangle.$

There are the desired inequalities, with $\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{5}\rangle$ in place of $\langle x_{i}\rangle$, and $10\varepsilon$ in
place of $\varepsilon$.

(1.1.2) we assume that exists nonzero projection $r$ such that
$\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle
r\rangle\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle$.

As in the part (1.1.1.1), as $A\in{\rm WTA}\Omega$, there exist a projection
$p_{4}\in A$, an element $g_{4}\in A$ with $0\leq g_{4}\leq 1$, and a $\rm
C^{*}$-subalgebra $D_{4}$ of $A$ with $g_{4}\in D_{4}$, $1_{D_{4}}=p_{4}$, and
$D_{4}\in\Omega$, by $(1)^{\prime}$, there exists a positive element $a^{6}\in
D_{4}$, such that

$1-(p_{4}-g_{4})\preceq r,$
$\|(p_{4}-g_{4})^{\frac{1}{2}}a^{{}^{\prime\prime}}(p_{4}-g_{4})^{\frac{1}{2}}-a^{6}\|<\varepsilon/3,$
and

$\|a^{{}^{\prime\prime}}-a^{6}-(1-(p_{4}-g_{4}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{4}-g_{4}))^{\frac{1}{2}}\|<\varepsilon.$

Also as in the part (1.1.1.1), we have
$\langle(a^{6}-2\varepsilon)_{+}\rangle\leq\langle a\rangle.$

Since $D_{4}$ is weakly $(m,n)$-divisible, $(a^{6}-2\varepsilon)_{+}\in
D_{4}$, there exist $x_{1}^{6},x_{2}^{6},$ $\cdots,x_{n}^{6}\in D_{4}$, such
that

$\langle x_{j}^{6}\rangle+\langle x_{j}^{6}\rangle+\cdots+\langle
x_{j}^{6}\rangle\leq\langle(a^{6}-2\varepsilon)_{+}\rangle$, and

$\langle(a^{6}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{6}\rangle$, where $\langle x_{j}^{6}\rangle$ repeats $m$ times.

Therefore, we have

$\langle y_{i}^{{}^{\prime}}\oplus x_{i}^{6}\rangle+\langle
y_{i}^{{}^{\prime}}\oplus x_{i}^{6}\rangle+\cdots+\langle
y_{i}^{{}^{\prime}}\oplus x_{i}^{6}\rangle$

$\leq\langle(a^{{}^{\prime}}-\varepsilon)_{+}\rangle+\langle(a^{6}-2\varepsilon)_{+}\rangle\leq\langle
a\rangle+\langle a\rangle$, where $\langle y_{i}^{{}^{\prime}}\oplus
x_{i}^{6}\rangle$ repeats $m$ times for $1\leq i\leq n$.

We also have

$\langle(a-10\varepsilon)_{+}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{6}-4\varepsilon)_{+}\rangle+\langle(1-(p_{4}-g_{4}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{4}-g_{4}))^{\frac{1}{2}}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{6}-4\varepsilon)_{+}\rangle+\langle(1-(p_{4}-g_{4}))\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{6}-4\varepsilon)_{+}\rangle+\langle
r\rangle$ $\leq\sum_{i=1}^{n}\langle y_{i}^{{}^{\prime}}\oplus
x_{i}^{6}\rangle.$

There are the desired inequalities, with$\langle y_{i}^{{}^{\prime}}\oplus
x_{i}^{6}\rangle$ in place of $\langle x_{i}\rangle$, and $10\varepsilon$ in
place of $\varepsilon$.

(1.2) If $(a^{{}^{\prime}}-3\varepsilon)_{+}$ is not Cuntz equivalent to a
projection, by Theorem 2.1 (3), then there is a nonzero positive element $d$
such that $\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle+\langle
d\rangle\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle$.

As in the part (1.1.1.1), since $A\in{\rm WTA}\Omega$, there exist a
projection $p_{5}\in A$, an element $g_{5}\in A$ with $0\leq g_{5}\leq 1$, and
a $\rm C^{*}$-subalgebra $D_{5}$ of $A$ with $g_{5}\in D_{5}$,
$1_{D_{5}}=p_{5}$, and $D_{5}\in\Omega$, by $(1)^{\prime}$, there exists a
positive element $a^{7}\in D_{5}$, such that

$1-(p_{5}-g_{5})\preceq d,$
$\|(p_{5}-g_{5})^{\frac{1}{2}}a^{{}^{\prime\prime}}(p_{5}-g_{5})^{\frac{1}{2}}-a^{7}\|<\varepsilon/3,$
and

$\|a^{{}^{\prime\prime}}-a^{7}-(1-(p_{5}-g_{5}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{5}-g_{5}))^{\frac{1}{2}}\|<\varepsilon.$

Also as in the part (1.1.1.1), we have
$\langle(a^{7}-2\varepsilon)_{+}\rangle\leq\langle a\rangle.$

Since $D_{5}$ is weakly $(m,n)$-divisible, $(a^{7}-2\varepsilon)_{+}\in
D_{5}$, there exist $x_{1}^{7},x_{2}^{7},$ $\cdots,x_{n}^{7}\in D_{5}$, such
that

$\langle x_{j}^{7}\rangle+\langle x_{j}^{7}\rangle+\cdots+\langle
x_{j}^{7}\rangle\leq\langle(a^{7}-2\varepsilon)_{+}\rangle$, and
$\langle(a^{7}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{7}\rangle$, where $\langle x_{j}^{7}\rangle$ repeats $m$ times.

Therefore, we have

$\langle x_{j}^{{}^{\prime}}\oplus x_{j}^{7}\rangle+\langle
x_{j}^{{}^{\prime}}\oplus x_{j}^{7}\rangle+\cdots+\langle
x_{j}^{{}^{\prime}}\oplus x_{j}^{7}\rangle$

$\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle+\langle(a^{7}-2\varepsilon)_{+}\rangle\leq\langle
a\rangle+\langle a\rangle$, where $\langle x_{j}^{{}^{\prime}}\oplus
x_{j}^{7}\rangle$ repeats $m$ times for $1\leq i\leq n$.

We also have

$\langle(a-10\varepsilon)_{+}\rangle$

$\leq\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle+\langle(a^{7}-4\varepsilon)_{+}\rangle+\langle(1-(p_{5}-g_{5}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{5}-g_{5}))^{\frac{1}{2}}\rangle$

$\leq\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle+\langle(a^{7}-4\varepsilon)_{+}\rangle+\langle(1-(p_{5}-g_{5}))\rangle$

$\leq\langle(a^{{}^{\prime}}-4\varepsilon)_{+}\rangle+\langle(a^{7}-4\varepsilon)_{+}\rangle+\langle
d\rangle$

$\leq\sum_{i=1}^{n}\langle x_{i}^{{}^{\prime}}\oplus x_{i}^{7}\rangle.$

There are the desired inequalities, with $\langle x_{i}^{{}^{\prime}}\oplus
x_{i}^{7}\rangle$ in place of $\langle x_{i}\rangle$, and $10\varepsilon$ in
place of $\varepsilon$.

Case(2) If $(a^{{}^{\prime}}-2\varepsilon)_{+}$ is not Cuntz equivalent to a
projection, by Theorem 2.1 (3), there is a nonzero positive element $d$ such
that $\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle
d\rangle\leq\langle(a^{{}^{\prime}}-2\varepsilon)_{+}\rangle$.

As in the part (1.1.1.1), since $A\in{\rm WTA}\Omega$, there exist a
projection $p_{6}\in A$, an element $g_{6}\in A$ with $0\leq g_{6}\leq 1$, and
a $\rm C^{*}$-subalgebra $D_{6}$ of $A$ with $g_{6}\in D_{6}$,
$1_{D_{6}}=p_{6}$, and $D_{6}\in\Omega$, by $(1)^{\prime}$, there exists a
positive element $a^{8}\in D_{6}$, such that

$1-(p_{6}-g_{6})\preceq d,$
$\|(p_{6}-g_{6})^{\frac{1}{2}}a^{{}^{\prime\prime}}(p_{6}-g_{6})^{\frac{1}{2}}-a^{8}\|<\varepsilon/3,$
and

$\|a^{{}^{\prime\prime}}-a^{8}-(1-(p_{6}-g_{6}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{6}-g_{6}))^{\frac{1}{2}}\|<\varepsilon.$

Also as in the part (1.1.1.1), we have
$\langle(a^{8}-2\varepsilon)_{+}\rangle\leq\langle a\rangle.$

Since $D_{6}$ is weakly $(m,n)$-divisible, $(a^{8}-2\varepsilon)_{+}\in
D_{6}$, there exist $x_{1}^{8},x_{2}^{8},$ $\cdots,x_{n}^{8}\in D_{6}$, such
that

$\langle x_{j}^{8}\rangle+\langle x_{j}^{8}\rangle+\cdots+\langle
x_{j}^{8}\rangle\leq\langle(a^{8}-2\varepsilon)_{+}\rangle$, and
$\langle(a^{8}-3\varepsilon)_{+}\rangle\leq\sum_{i=1}^{n}\langle
x_{i}^{8}\rangle$, where $\langle x_{j}^{8}\rangle$ repeats $m$ times.

Therefore, we have

$\langle y_{i}^{{}^{\prime}}\oplus x_{i}^{8}\rangle+\langle
y_{i}^{{}^{\prime}}\oplus x_{i}^{8}\rangle+\cdots+\langle
y_{i}^{{}^{\prime}}\oplus x_{i}^{8}\rangle$

$\leq\langle(a^{{}^{\prime}}-\varepsilon)_{+}\rangle+\langle(a^{8}-2\varepsilon)_{+}\rangle\leq\langle
a\rangle+\langle a\rangle$, where $\langle y_{i}^{{}^{\prime}}\oplus
x_{i}^{8}\rangle$ repeats $m$ times for $1\leq i\leq n$.

We also have

$\langle(a-10\varepsilon)_{+}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{8}-4\varepsilon)_{+}\rangle+\langle(1-(p_{6}-g_{6}))^{\frac{1}{2}}a^{{}^{\prime\prime}}(1-(p_{6}-g_{6}))^{\frac{1}{2}}\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{8}-4\varepsilon)_{+}\rangle+\langle(1-(p_{6}-g_{6}))\rangle$

$\leq\langle(a^{{}^{\prime}}-3\varepsilon)_{+}\rangle+\langle(a^{8}-4\varepsilon)_{+}\rangle+\langle
d\rangle$ $\leq\sum_{i=1}^{n}\langle y_{i}^{{}^{\prime}}\oplus
x_{i}^{8}\rangle.$

There are the desired inequalities, with $\langle y_{i}^{{}^{\prime}}\oplus
x_{i}^{8}\rangle$ in place of $\langle x_{i}\rangle$, and $10\varepsilon$ in
place of $\varepsilon$. ∎

The following corollary were obtained by Fan, Fang, and Zhao in [17].

######  Corollary 3.8.

Let $A$ be a unital simple stably finite separable ${\rm C^{*}}$-algebra. Let
$B\subseteq A$ be a centrally large subalgebra of $A$ such that $B$ is weakly
$(m,n)$-divisible. Then $A$ is secondly weakly $(m,n)$-divisible.

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# Engagement Decision Support for Beyond Visual Range Air Combat

Joao P. A. Dantas1, Andre N. Costa1, Diego Geraldo1, Marcos R. O. A. Maximo2,
and Takashi Yoneyama3 1Joao P. A. Dantas, Andre N. Costa, and Diego Geraldo
are with Institute for Advanced Studies, Sao Jose dos Campos, SP, 12.288-001,
Brazil<EMAIL_ADDRESS>R. O. A. Maximo is
with Autonomous Computational System Lab (LAB-SCA), Computer Science Division,
Aeronautics Institute of Technology, Sao Jose dos Campos, SP, 12228-900,
Brazil<EMAIL_ADDRESS>Yoneyama is with Electronic Engineering
Division, Aeronautics Institute of Technology, Sao Jose dos Campos, SP,
12228-900, Brazil<EMAIL_ADDRESS>
###### Abstract

This work aims to provide an engagement decision support tool for Beyond
Visual Range (BVR) air combat in the context of Defensive Counter Air (DCA)
missions. In BVR air combat, engagement decision refers to the choice of the
moment the pilot engages a target by assuming an offensive stance and
executing corresponding maneuvers. To model this decision, we use the
Brazilian Air Force’s Aerospace Simulation Environment (Ambiente de Simulação
Aeroespacial - ASA in Portuguese), which generated 3,729 constructive
simulations lasting 12 minutes each and a total of 10,316 engagements. We
analyzed all samples by an operational metric called the DCA index, which
represents, based on the experience of subject matter experts, the degree of
success in this type of mission. This metric considers the distances of the
aircraft of the same team and the opposite team, the point of Combat Air
Patrol, and the number of missiles used. By defining the engagement status
right before it starts and the average of the DCA index throughout the
engagement, we create a supervised learning model to determine the quality of
a new engagement. An algorithm based on decision trees, working with the
XGBoost library, provides a regression model to predict the DCA index with a
coefficient of determination close to 0.8 and a Root Mean Square Error of 0.05
that can furnish parameters to the BVR pilot to decide whether or not to
engage. Thus, using data obtained through simulations, this work contributes
by building a decision support system based on machine learning for BVR air
combat.

##  I INTRODUCTION

Air combat may occur in two primary forms: Within Visual Range (WVR) and
Beyond Visual Range (BVR) [1], with the latter being recently more developed
in the operational context, due to more extensive availability of more
advanced weapons and sensors [2]. Notice that, even though modern air combat
may still end WVR, through a series of complex decisions and maneuvers, it
usually begins BVR, which frequently is the most critical phase of the combat
since it may provide advantages and drawbacks for succeeding phases [3]. There
is no clear definition of the distance to differentiate these two forms of air
combat since this may be subject to the conditions in which the air combat
happens.

BVR conditions force the pilots to rely more on a series of systems to compose
their situational awareness, allowing them to make tactical decisions during
the combat, such as whether to fire a missile or not. Especially in BVR
combat, the missile launches and the circumstances around these events are
critical [4], since these weapons are the main form to engage the opponent.
Since BVR combat is rarely observable in practice, with low availability of
historical data, much of the assessment of its possibilities must be done
through simulation [5]. It is also true due to the high costs of flying, air
space regulations, and limited availability of platforms representative of
those used by opposing forces [6].

This work contributes by developing a decision support tool for pilots in BVR
combat situations based on simulated data, with a particular focus on solving
the problem of deciding when to engage a specific enemy aircraft, which is
commonly based solely on pilot experience. There are several previous
approaches that relate to this decision, mainly providing different forms of
modeling pilot behavior. Many of them use game theory to model air combat [7,
8, 9, 10]. Other approaches found in the literature are as follows: Bayesian
Networks [11, 12, 13, 14], fuzzy logic [15, 16], agent-based modeling [17],
influence diagrams [18], reinforcement learning [19, 20, 21, 22], artificial
neural networks [23, 24], evolutionary algorithms [25, 26], minimax method
[27], and behavior trees [28].

Among all of these approaches, Ha _et al._ (2018) [10] is the one that more
directly focuses on estimating the firing moment, which is done through a
probabilistic function of the target’s evasive maneuverability, the missile’s
speed on the final approach, and the accuracy of the target’s information to
guide the missile. However, as with many of the other cited methods, the
methodology proposed in [10] has not been tested in simulations with a higher
degree of fidelity concerning an actual BVR air combat.

As an alternative, our work uses supervised machine learning models based on
decision trees, using the XGBoost library [29], to provide pilots with
parameters that improve their situational awareness in air combat from data
collected from simulations of operational scenarios, allowing them to better
decide when to engage a target. Compared to most of the approaches, this study
was conducted through higher fidelity simulations, providing systems and
subsystems that resemble their natural counterparts through our simulation
environment, such as 6 degrees of freedom (6DOF) multi-role combat aircraft,
electronic warfare (EW) devices, datalink communications, and active radar-
guided missiles.

This study is, therefore, developed around four main subjects: i)
determination of BVR air combat scenarios that will be analyzed, ii) carrying
out simulations based on the chosen scenarios in the agents’ configurations
are varied, iii) collecting and analyzing the data generated in the
simulations, and iv) using machine learning techniques on this data to improve
the pilot’s situational awareness by providing information about the analyzed
air combat, fulfilling the role of a decision support system.

The remainder of this paper is organized as follows. In section 2, the main
characteristics of the aircraft in the BVR air combat are described. We also
explain the Defensive Counter Air (DCA) index, the operational metric used to
perform the engagement analysis. Besides, the sampling process of the
simulation inputs is discussed. In section 3, we show the methodology used to
create the solution approach. Section 4 investigates the results and analysis
concerning the exploratory data analysis and model results. Finally, in
section 5, we present the conclusions and future works.

##  II PROPOSAL DESCRIPTION

This section shows the Fighter Agent model, describing the simulation agent’s
reasoning that we propose. Next, the DCA index, the operational metric that we
created to evaluate the air combat engagements, is presented. Finally, we
discuss our process of sampling simulation input parameters.

###  II-A Fighter Agent

The Institute for Advanced Studies from the Brazilian Air Force develops the
Aerospace Simulation Environment (_Ambiente de Simulação Aeroespacial - ASA_
in Portuguese) to provide a computational solution that enables the simulation
of operational scenarios, allowing users to establish scenarios, parameters,
and command decisions to support the development of tactics, techniques, and
procedures. Since there is so much variability in the nature of military
scenarios, it may be hard to define what the scenarios of interest are. Thus,
the development of ASA is not limited to predefined layouts; instead, it poses
itself as a flexible solution that may be tailored to the user’s needs. Its
modularity contributes to this flexibility since one can configure components
(models) in diverse ways and combine them freely, which enables the creation
of the most varied scenarios [30].

ASA is a custom-made, object-oriented (C++), and high-fidelity environment
that generates the simulations and the data to analyze the problem. The
simulation concept addressed in this context is one in which the scenario
elements are represented as agents capable of making decisions based on
artificial intelligence models or arbitrary rules previously stipulated.
Simulations of this nature, termed constructive, can be used in the decision-
making process, for example, to predict possible outcomes of engagements
between opposing forces and assist in the definition of lines of action.

The agent’s actions are managed by a set of rules that describe a pilot’s main
tactical behaviors during a BVR combat. This set of rules is modeled as a
Finite State Machine (FSM), or finite automaton, which is a mathematical
computer model that, at any time, can be in only one state of a limited set of
states [31]. Each state is one of the possible tactics that can be taken
during a BVR combat. The following behaviors will be analyzed to better
understand the presented problem: Combat Air Patrol (CAP), Commit, Abort, and
Break. Fig. 1 shows the FSM that manages all the agent’s tactics.

Figure 1: FSM of agent tactics.

The CAP tactic consists of performing a flight pattern, describing an orbit,
that can be any tactical maneuver, such as a circle or an oval, around a
specific position called CAP point, with a defined heading and direction
(clockwise or counterclockwise). The Commit tactic consists of the agent
engaging a target detected by its radar or shared by data link by its allies.
In the Abort tactic, the agent performs a defensive maneuver to move away from
his priority threat, which is also the priority target in many situations.
Finally, when the agent’s sensors detect a missile threat fired in its
direction, it performs the Break tactic. This tactic consists of a sudden
defensive maneuver, describing a curve and a dive with great acceleration.

At the engagement moment, the FSM assumes the Commit behavior and will keep
performing offensive maneuvers until the time it is desirable within combat.
The offense and vulnerability indices are essential variables that guide the
change from one state to another within the FSM. Then, when the agent needs to
do a defensive movement, it assumes Break or Abort. Therefore, the engagement
time in a simulation is defined as the time between the first Commit carried
by the agent and its first Break or Abort. We created an operational index
that determines the quality of the actions taken by the agent in a BVR
scenario, which was calculated during the engagement period and can be
extended to any moment in the simulation. This metric is referred to as the
DCA Index, which will be detailed in the following subsection.

###  II-B DCA Index

We defined the index as a probability of success, ranging from 0% to 100%, for
BVR combat on DCA missions whose objective is to establish a CAP. These
missions have the goal of defending a point of interest, which is done by
ensuring that the opposing aircraft are kept far away from it. In addition, it
aims to do that while launching the least number of missiles possible, which
is interesting both from economic and operational standpoints. Furthermore,
from the doctrine perspective, one may consider it good practice for the
defending aircraft to stay close to its CAP point since it is easier to employ
tactics to defend the point of interest.

From these considerations with respect to the DCA mission context, we defined
three basic principles (depicted in Fig. 2) for the elaboration of the DCA
index:

  1. 1.

Minimize the number of missile launched in the mission:
($m_{total}-m_{avail}$).

  2. 2.

Minimize the reference distance from its CAP point: $D(r,CAP)$.

  3. 3.

Maximize the distance of each enemy ($e_{n}$) to the CAP point:
$D(e_{n},CAP)$.

Figure 2: Representation of the factors that form the DCA index.

Firstly, at the beginning of the engagement, each aircraft has a fixed number
of missiles ($n_{total}$), which at that moment is also the number of missiles
available ($n_{avail}$). When the aircraft launches a missile, $n_{avail}$
decreases to keep track of the currently available missiles. The ratio between
$n_{avail}$ and $n_{total}$ is one of the factors of the DCA index, so that,
when maximizing this ratio, $n_{total}-n_{avail}$ is minimized.

Secondly, regarding minimizing the distance between the reference aircraft and
the CAP point, i.e., $D(r,CAP)$, we considered that the decay of this effect
is not linear since, as the distance increases, its influence becomes much
less relevant to fulfilling the mission goals. Therefore, we chose a sigmoid
function to generate such decay to encompass this non-linearity. We defined
the sigmoid limits, considering operational experience with respect to the
maximum range of the available missile, as 8,000 meters ($x_{99\%,r}$) for the
99% output value, which corresponds to $y_{99\%,r}\approx 4,5951$, and as
12,000 ($x_{1\%,r}$) meters for the 1% output value, which stands for
$y_{1\%,r}\approx-4,5951$. These limits must be used in a linear interpolation
(1) to convert the current distance value ($D_{r}$) to be input ($d_{r}$) in
the sigmoid equation (2).

$d_{i}=\frac{(y_{99\%,i}-y_{1\%,i})}{(x_{99\%,i}-x_{1\%,i})}\cdot[D(i,CAP)-x_{1\%,i}]+y_{1\%,i}$
(1)

where:  $i$ ${}={}$ $r$ (reference) or $e_{n}$ (enemy) $D_{i}$ ${}={}$
measured distance from the CAP point $d_{i}$ ${}={}$ interpolated distance for
sigmoid input

Lastly, when considering the opposing aircraft distances ($D(e_{n},CAP)$), for
all $N$ enemies, we used a similar sigmoid function, but with opposite
characteristics since the idea was to increase the enemies’ influence when
they were closer to the CAP point. Therefore, the sigmoid limits for this
factor were instead 12,000 ($x_{99\%,e}$) meters for the 99% output value,
which corresponds to $y_{99\%,e}\approx 4,5951$, and 8,000 ($x_{99\%,e}$)
meters for the 1% output value, which stands for $y_{1\%,e}\approx-4,5951$.
Applying the enemy sigmoid limits in the interpolation equation (1), we are
able to convert the measured distance value $D(e_{n},CAP)$ to be input
($d_{e_{n}}$) in (2) for each of the enemy aircraft.

With the composition of the previously calculated factors, it is possible to
obtain the DCA index’s final calculation. The index factors have coefficients
to allow prioritization of the three principles presented, as seen in (2). We
defined the weights ($w_{1}=0.2$, $w_{2}=0.4$, and $w_{3}=0.4$) for each
factor of the DCA index ($I_{DCA}$) based on the operational knowledge from
subject matter experts.

$\displaystyle I_{DCA}=$
$\displaystyle\;w_{1}\cdot\frac{m_{avail}}{m_{total}}+w_{2}\cdot\frac{1}{1+\exp(-d_{r})}$
(2)
$\displaystyle+w_{3}\cdot\frac{1}{N}\sum_{n=1}^{N}\frac{1}{1+\exp(-d_{e_{n}})}$

###  II-C Sampling of Simulation Input Parameters

Latin Hypercube Sampling (LHS) is a method that can be used to produce, in a
distributed way, the set of input values to be used in the simulations
according to the desired intervals [32]. This statistical method generates a
random sample of parameter values from distribution in multiple dimensions.
LHS consists of subdividing the sample universe into several disjoint subsets
and extracting a representative element for each subsets, chosen at random.

The generated and stored simulations used in the supervised learning model
were carried out in packages (batches). Thus, for the same type of scenario,
it was possible to vary some parameters at the beginning of the simulation,
which may lead to different outcomes. The simulations parameters changed
during the sampling were: (a) latitude and longitude, determining the initial
positions of the agents around the CAP points (adopting fixed CAP positions),
(b) the flight level blocks to determine their altitudes, (c) the commit
distance (the minimum distance that an agent is from a possible target that
allows it to leave the CAP tactic and commit), (d) the thresholds of the
offense and vulnerability indices before and after firing a missile (represent
the level of risk acceptance that the agent is willing to withstand), (e) the
shot philosophy (orientation defined during mission planning, before the
flight, referring to the moment when, within the Weapon Engagement Zone (WEZ),
which is an estimation of the missile maximum launch range [33], the agent
must launch a missile), (f) the shot distance (the minimum distance that an
agent is from a possible target that allows, during an engagement, fire a
missile, i.e., the WEZ), and (g) the presence or absence on the aircraft of a
Radar Warning Receiver (RWR), an EW system that detects electromagnetic
emissions from opposing radar systems.

Using the LHS algorithm, 3,729 constructive simulations were generated in ASA,
and a total of 10,316 engagements were observed. Each simulation corresponds
to a 12-minute scenario executed three times faster than real-time, lasting
approximately ten days in total.

The scenario consists of two opposing formations with two aircraft each, which
are initially approaching, disengaged, and outside the radar range of each
other. Their main goal is to establish a CAP at the same CAP point, invariably
leading to a confrontation. When they enter the limits of the opponents’
radars, the engagement phase begins. In the modeling proposed in this work,
each aircraft is equipped with four of the same type of medium-range missile,
i.e., up to 40 nautical miles in range.

##  III METHODOLOGY

After sampling the variables, respecting the intervals chosen for each one, we
executed the simulations through ASA. The engagement events between the four
agents are extracted with the data generated from these simulations. The
average DCA index is calculated for all engagements extracted from the
interval under analysis, generating the output variable that we intend to
predict later for new samples. With the input data of the simulations and the
output variable already defined, we build a supervised machine learning model
based on eXtreme Gradient Boosting (XGBoost), predicting the average value of
the agents’ DCA index in future engagements. XGBoost represents a class of
algorithms based on Decision Trees with Gradient Boosting [34]. Its
performance is analyzed with the test dataset after the model’s training
process is completed. This section discusses the input and output model
variables, preprocessing procedures, hyperparameters tunning, evaluation
metrics, and cross-validation processes.

###  III-A Model Input and Target Variables

The main variables that coordinate the simulation of a BVR air combat were
analyzed, and seventeen input variables were determined to be the most
important to define the progress of the simulations based on the described
scenarios. In addition, there are categorical and numerical variables with
different ranges of coverage, and the definition of these sampling intervals
was made based on the operational knowledge of BVR pilots and combat
specialists. Next, in Table I, the description of each of the input variables
of the simulations is carried out, presenting its unit when the variable is
not dimensionless.

TABLE I: Variables at the beginning of the engagement. Parameter |  Description
---|---
distance [m] |  Distance between the reference and the target
aspect [deg] |  Angle between the longitudinal axis of the target (projected rearward) and the line-of-sight to the reference
delta_head [deg] |  Angle between the longitudinal axis of both aircraft
delta_alt [m] |  Difference of altitude between the reference and the target
delta_vel [kn] |  Difference of absolute velocity between the reference and the target
wez_max_o2t [m] |  Maximum range of the reference’s weapon (non-maneuverable target)
wez_nez_o2t [m] |  No-escape zone range of the reference’s weapon (target performing high performance maneuver)
wez_max_t2o [m] |  Estimated maximum range of the target’s weapon (non-maneuverable reference)
wez_nez_t2o [m] |  No-escape zone range of the target’s weapon (reference performing high performance maneuver)
vul_thr_bef_shot |  Level of risk acceptance before shooting
vul_thr_aft_shot |  Level of risk acceptance after shooting
shot_point |  Missile firing point between the maximum range and the no-escape zone range of the reference
rwr_warning |  Boolean indicating whether the aircraft is equipped with an active RWR
hp_tgt_off |  High priority target offense index of the reference
hp_thr_vul |  High priority threat vulnerability index of the aircraft that is threatening the reference
own_shot_phi |  Reference shot philosophy
enemy_shot_phi |  Estimated enemy’s shot philosophy

Concerning the target parameter, the DCA index will be averaged between the
start of the agent’s Commit maneuver and the beginning of either Break or
Abort maneuvers since both will make the agent disengage. Therefore, given a
sequence of input parameters that define the agent’s state, the model must
predict the average value of the DCA index in that interval, improving the
agent’s situational awareness.

###  III-B Data Preprocessing

Unlike what is done in artificial neural networks, there is no need to carry
out data normalization procedures to employ the XGBoost algorithm since it is
based on decision trees. Thus, using this type of learning method, the model
benefits from one of the significant advantages of these trees in artificial
intelligence problems related to the low amount of preprocessing required for
it to be applied. It is necessary to transform the model’s categorical
variables into numerical ones to be appropriately processed. For this purpose,
two preprocessing steps will be performed on the data: Label Encoding and One-
Hot Encoding [35]. Also, feature engineering will be carried out to facilitate
the training process of the proposed regression model.

###  III-C Hyperparameters Tuning

GridSearch is a hyperparameter adjustment process to determine the ideal
values for a given model, based on searching throughout a grid [36]. The
performance of the entire model is based on the specified hyperparameter
values. Some functions have been implemented, such as the GridSearchCV of the
sklearn library, to automate finding the best of these values for the model.
We performed an adjustment of the XGBoost model hyperparameters based on a
variation of the library default values as observed in Table II.

TABLE II: GridSearch parameters for the prediction model. Parameters | Values
---|---
n_estimators | [100, 1000, 5000]
learning_rate | [2, 3, 6, 10, 15, 20]
max_depth | [0.1, 0.01, 0.001]
gamma | [0.0, 0.1, 0.2, 0.3, 0.4, 0.5]
subsample | [0.6, 0.7, 0.8, 0.9]
colsample_bytree | [0.6, 0.7, 0.8, 0.9]
reg_alpha | [0.001, 0.01, 0.1, 1, 10, 100]
min_child_weight | [1,3,5,7,9,10,13,15]

###  III-D Metrics

The Root-Mean-Square Error (RMSE) is used to measure the differences between
the values predicted by a model or an estimator concerning the actual values,
providing the average magnitude of the error. As the errors are squared, the
RMSE places a relatively high weight on significant errors, which means that
RMSE should be most useful when large errors are particularly undesirable. In
addition, the advantage of using the RMSE is that it has the same magnitude as
the target variable, which helps in the interpretation of the average of the
model errors found. In the analysis of the predictive models carried out in
this work, the coefficient of determination ($R^{2}$) will also be used to
evaluate the best architectures of the supervised machine learning models for
BVR combat modeling. The use of $R^{2}$ is well established in classical
regression analysis and it generally describes how the model is adapted to
reality when making predictions.

###  III-E Cross-Validation

After performing a dataset train-validation-test split, allocating 80% for the
training and validation and 20% for the testing, we propose to conduct cross-
validation to evaluate the model’s ability to predict new data that was not
used in the estimate with the benefits of using the whole train dataset for
training and validation and to highlight problems such as overfitting [37]. We
employ 10-fold cross-validation to address this problem after considering the
trade-off between processing time and the generalization of the model results.

##  IV RESULTS AND ANALYSIS

This section presents a whole dataset exploratory data analysis to provide an
initial understanding of the variables in the model, followed by 10-fold
cross-validation results.

###  IV-A Exploratory Data Analysis

Exploratory data analysis began with a check of the main descriptive
statistics of the model’s input and output data. The input variables of the
model follow a uniform distribution since they were sampled using LHS. The
model’s output variable, the DCA index, is a probability, and in this case, it
ranges from a minimum value of 0.21 to a maximum value of 0.99. with a mean of
0.53 and a standard deviation of 0.12. Moreover, the mean and the median are
almost the same (0.53 and 0.51), indicating a low number of outliers for this
variable at the top of the distribution. A histogram and a boxplot were
generated to visualize the distribution, as shown in Fig. 3. The data of the
target variable follows an approximately normal distribution around the
average value.

Regarding the aspect and heading difference, these two angular variables are
transformed into four numerical variables to calculate their respective sine
and cosine values. Note that the variables referring to WEZ have minimum
values of $-1$. These values are model adjustments for when it is not possible
to estimate the WEZ.

Figure 3: Histogram and boxplot of the target variable.

###  IV-B Model Results

The means of all 10-fold metrics, namely RMSE and $R^{2}$, used to evaluate
the best model at the end of the grid search training process after performing
the cross-validation are, respectively, $0.0543$ and 0.8020, while their
standard deviations are 0.0009 and 0.0077. The coefficient of determination is
approximately 80%, and an RMSE, which penalizes the effects of outliers, is
close to 0.05. Considering that this is a regression problem and that we are
trying to predict the DCA index, which indicates a probability of success in
this type of mission, the results are satisfactory for this type of problem
since they would be making predictions with errors in the range of 5% with a
practically instantaneous inference time, which is desirable for a real-time
application.

##  V Conclusions and Future Work

This work presented a supervised machine learning model through the XGBoost
library to develop an engagement decision support tool for BVR air combat in
DCA missions. The model represented some of the primary dynamics of the BVR
combat, allowing the analyst to evaluate the parameters that influence this
combat. Additionally, we analyze in advance the performance of an engagement
made by a pilot in air combat through the agent’s DCA index, predicting the
outcome of this confrontation. This kind of prediction may be used as an
innovative decision support system for the pilots in this air combat modality
concerning whether to engage an opponent or not. Although the index does not
inform which is the better action to take, it measures the performance of the
actions taken by the agent through the proposed modelings.

The modeling considered the characteristics of the aircraft and their
armaments, along with the beliefs about the opposing aircraft. In addition, we
also used the shot philosophy for the aircraft and the pilot’s level of risk
aversion. The model showed an $R^{2}$ of 0.802 and an RMSE of 0.054. Assuming
the average values of the DCA index as 0.53 and standard deviation of 0.12,
the results showed relatively consistent values and good predictive power of
the DCA index. With this degree of confidence in the model, it is possible to
predict future pilot’s conditions, even with a few samples. The regression
model could calculate the average values of the DCA index in each engagement
in a coherent manner, providing quick answers to the results of this air
combat phase that could provide the pilot with an improvement in his
situational awareness in real-time. Thus, through simulation data, it is
possible to improve the employment of the best operational tactics for each
situation in the complex context of BVR air combat. Furthermore, it
contributes by avoiding the incorrect and careless use of weapons, improving
DCA mission effectiveness. Finally, since the pilot’s survival is a
determining factor, it could decrease the number of friendly aircraft lost in
real-life BVR air combat.

Future work should move towards using a more significant number of simulations
since we only analyzed 10,316 engagement cases due to the substantial
computational cost to generate them. In addition, the search for more
variables to define the agent state at the beginning of the engagement, or to
do a feature engineering to generate new variables from the existing ones,
could improve the performance of the proposed model. Also, comparing the
results of XGBoost with other regression algorithms would be interesting to
understand which supervised learning method best suits this type of problem.
Finally, the conception of other operational metrics, like the DCA index, or
the combination of several of them, to assess the level of performance in an
engagement could contribute to bringing more information to the model.

## Acknowledgments

This work was supported by Finep (Reference nº 2824/20). Takashi Yoneyama is
partially funded by CNPq – National Research Council of Brazil through the
grant 304134/2-18-0.

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