arXiv subjects
Publications and source records attributed to Qingzhai Fan.
Let $A$ be an infinite-dimensional unital simple C*-algebra such that A has $μ$-$oz{\rm LLP}$ property for any $μ\in(0,1)$. Let $B\subseteq A$ be a centrally large subalgebra of $A$ such that $B$ has tracial topological rank zero. Then $A$ has tracial topological rank zero.
We show that the following divisible properties of the ${\rm C^*}$-algebras in a class $\mathcal{P}$ are inherited by simple unital ${\rm C^*}$-algebras in the class of asymptotically tracially in $\mathcal{P}$: $(1)$ $m$-almost divisible, $(2)$ weakly $(m, n)$-divisible.
We show that the following properties of unital ${\rm C^*}$-algebra in a class of $Ω$ are preserved by unital simple ${\rm C^*}$-algebra in the class of $\rm WTAΩ$: $(1)$ uniform property $Γ$, $(2)$ a certain type of tracial nuclear dimension at most $n$, $(3)$ weakly $(m, n)$-divisible.
Let $Ω$ be a class of ${\rm C^*}$-algebras. In this paper, we study a class of not necessarily unital generalized tracial approximation ${\rm C^*}$-algebras, and the class of simple ${\rm C^*}$-algebras which can be generally tracially approximated by ${\rm C^*}$-algebras in $Ω$, denoted by ${\rm gTA}Ω$. Let $Ω$ be a class of unital ${\rm C^*}$-algebras and let $A$ be a simple unital ${\rm C^*}$-algebra. Then $A\in {\rm gTA}Ω$, if, and only if, $A\in {\rm WTA}Ω$ (where ${\rm TA}Ω$ is the class of weakly tracially approximable unital ${\rm C^*}$-algebras introduced by Elliott, Fan, and Fang).Consider the class of ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or are of tracial nuclear dimension at most $n$, or are $m$-almost divisible, or have the property $\rm SP$). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, is weakly ($n, m$)-almost divisible, has the property $\rm SP$) for any simple ${\rm C^*}$-algebra $A$ in the corresponding class of generalized tracial approximation ${\rm C^*}$-algebras.
Let $Ω$ be a class of unital $\rm C^{*}$-algebras. The class of ${\rm C^*}$-algebras which are asymptotical tracially in $Ω$, denoted by ${\rm AT}Ω$. In this paper, we will show that the following class of ${\rm C^*}$-algebras in the class $Ω$ are inherited by simple unital ${\rm C^*}$-algebras in the class $\rm TΩ$ $(1)$ the class of real rank zero ${\rm C^*}$-algebras, $(2)$ the class of ${\rm C^*}$-algebras with the radius of comparison $n$, and $(3)$ the class of $\rm ATΩ$. As an application, let $Ω$ be a class of unital ${\rm C^*}$-algebras which have generalized tracial rank at most one (or has tracial topological rank zero, or has tracial topological rank one). Let $A$ be a unital separable simple ${\rm C^*}$-algebra such that $A\in \rm ATΩ$, then $A$ has generalized tracial rank at most one (or has tracial topological rank zero, or has tracial topological rank one).
Let $Ω$ 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 $Ω$. Then ${\rm T^2dim_{nuc}}(A)\leq n$ (or ${\rm Tdim_{nuc}}(A)\leq n$). As an application, let $A$ be an infinite dimensional simple separable amenable unital ${\rm C^*}$-algebra with ${\rm T^2dim_{nuc}}(A)\leq n$ (or ${\rm Tdim_{nuc}}(A)\leq n$). Suppose that $α:G\to {\rm Aut}(A)$ is an action of a finite group $G$ on $A$ which has the tracial Rokhlin property. Then ${\rm T^2dim_{nuc}}({{\rm C^*}(G, A,α)})\leq n$ (or ${\rm Tdim_{nuc}}$ $({{\rm C^*}(G, A,α)})\leq n$).
In this paper, we introduce some classes of generalized tracial approximation ${\rm C^*}$-algebras. Consider the class of unital ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (or have tracial nuclear dimension at most $n$, or have the property $\rm SP$, or are $m$-almost divisible). Then $A$ is tracially $\mathcal{Z}$-absorbing (respectively, has tracial nuclear dimension at most $n$, has the property $\rm SP$, is weakly ($n, m$)-almost divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the corresponding class of generalized tracial approximation ${\rm C^*}$-algebras. As an application, let $A$ be an infinite-dimensional unital simple ${\rm C^*}$-algebra, and let $B$ be a centrally large subalgebra of $A$. If $B$ is tracially $\mathcal{Z}$-absorbing, then $A$ is tracially $\mathcal{Z}$-absorbing. This result was obtained by Archey, Buck, and Phillips in \cite{AJN}.
In this paper, we introduce a class of non-unital tracial approximation ${\rm C^*}$-algebras. Consider the class of ${\rm C^*}$-algebras which are tracially $\mathcal{Z}$-absorbing (in the sense of Amint, Golestani, Jamali, Phillips's simple tracially $\mathcal{Z}$-absorbing or Castillejos, Li, Szabvo's tracial $\mathcal{Z}$-stability). Then $A$ is tracially $\mathcal{Z}$-absorbing for any simple ${\rm C^*}$-algebra $A$ in the corresponding class of non-unital tracial approximation ${\rm C^*}$-algebras.
In this paper, we introduce a class of generalized tracial approximation ${\rm C^*}$-algebras. Let $\mathcal{P}$ be a class of unital ${\rm C^*}$-algebras which have tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, $m$-almost divisible, weakly $(m, n)$-divisible). Then $A$ has tracially $\mathcal{Z}$-absorbing (tracial nuclear dimension at most $n$, $\rm SP$ property, weakly $m$-almost divisible, secondly weakly $(m, n)$-divisible) for any simple unital ${\rm C^*}$-algebra $A$ in the class of this generalized tracial approximation ${\rm C^*}$-algebras. As an application, Let $A$ be an infinite dimensional unital simple ${\rm C^*}$-algebra, and let $B$ be a centrally large subalgebra of $A$. If $B$ is tracially $\mathcal{Z}$-absorbing, then $A$ is tracially $\mathcal{Z}$-absorbing. This result was obtained by Archey, Buck and Phillips in \cite{AJN}.
We show that the following properties of the C*-algebras in a class $\mathcal{P}$ are inherited by simple unital ${\rm C^*}$-algebras in the class of asymptotically tracially in $\mathcal{P}$: $(1)$ $β$-comparison (in the sense of Kirchberg and Rørdam), $(2)$ $n$-comparison (in the sense of Winter).
In this paper, we give two properties of C*-algebra that could be deduced from the properties of its large subalgebra. Let A be an infinite dimensional simple unital C*-algebra and let B be a centrally large subalgebra of A, we prove that A has real rank zero if B has real rank zero. If A is stablely fnite in addition, B is a large subalgebra of A, we prove that B has local weak comparison if A has local weak comparison, and A has local weak comparison if M2(B) has local weak comparison. As a consequence, we show that A has weak comparison if and only if B has weak comparison. These results could be used to study some properties of C*-algebra from its large subalgebra or centrally large subalgebra.