Searcharxiv⌕ Search

arXiv subjects

Xiaochun Fang

Publications and source records attributed to Xiaochun Fang.

17 recordsLinked to original sources
arXiv2026

Rokhlin dimension for actions of residually compact groups

We introduce the concept of Rokhlin dimension for actions of residually compact groups on C*-algebras, which extends and unifies previous notions for actions of compact groups, residually finite groups and the reals. We then demonstrate that finite nuclear dimension (respectively, absorption of a strongly self-absorbing C*-algebra) is preserved under the formation of crossed products by residually compact group actions with finite Rokhlin dimension (respectively, finite Rokhlin dimension with commuting towers). Furthermore, if second countable residually compact group contains a non-open cocompact closed subgroup, then crossed products arising from actions with finite Rokhlin dimension are stable. Finally, we study the relationship between the tube dimension of a topological dynamical system and the Rokhlin dimension of the induced C*-dynamical system.

math.OA↗
arXiv2026

Crossed products by compact group actions with the weak tracial Rokhlin property

In this paper, we introduce compact group actions with the weak tracial Rokhlin property. This concept simultaneously generalizes finite group actions with the weak tracial Rokhlin property and compact group actions with the tracial Rokhlin property (in the sense of the Elliott program). Under this framework, we prove that simplicity, pure infiniteness, tracial $\mathcal{Z}$-stability and the combination of nuclearity and $\mathcal{Z}$-stability can be transferred from the original algebra to the crossed product. We also show that the radius of comparison of the fixed point algebra does not exceed that of the original algebra. Furthermore, we discuss the relationship between our definition and natural generalization of the finite group case in non-Elliott program settings. Finally, we provide a nontrivial example of a compact group action with the weak tracial Rokhlin property with comparison: an action of $(S_2)^\mathbb{N}$ on the Jiang-Su algebra $\mathcal{Z}$. Since $\mathcal{Z}$ contains no nontrivial projections, this action does not possess the tracial Rokhlin property.

math.OA↗
arXiv2025

Uniform property $Γ$ for Crossed products by group actions with the Rokhlin-type properties

In this paper, let $A$ be a unital separable simple infinite dimensional C*-algebra which has uniform property $Γ$. Let $α\colon G\to \mathrm{Aut}(A)$ be an action of a finite group which has the weak tracial Rokhlin property. Then we prove that the crossed product $A\rtimes_αG$ and fixed point algebra $A^α$ have uniform property $Γ$. Let $α\colon G\to \mathrm{Aut}(A)$ be an action of a second-countable compact group which has the tracial Rokhlin property with comparison. Then we prove that the crossed product $A\rtimes_αG$ and fixed point algebra $A^α$ have uniform property $Γ$.

math.OA↗
arXiv2024

Stable rank for crossed products by finite group actions with the weak tracial Rokhlin property

Let $A$ be an infinite-dimensional stably finite simple unital C*-algebra, let $G$ be a finite group, and let $α\colon G\rightarrow \mathrm{Aut}(A)$ be an action of $G$ on $A$ which has the weak tracial Rokhlin property. We prove that if $A$ has property (TM), then the crossed product $A\rtimes_αG$ has property (TM). As a corollary, if $A$ is an infinite-dimensional separable simple unital C*-algebra which has stable rank one and strict comparison, $α\colon G\rightarrow \mathrm{Aut}(A)$ is an action of a finite group $G$ on $A$ with the weak tracial Rokhlin property, then $A\rtimes_αG$ has stable rank one.

math.OA↗
arXiv2023

Some Permanence properties for crossed products by compact group actions with the tracial Rokhlin property

In this paper, we give some properties of the fixed point algebra and the crossed product of a unital separable simple infinite dimensional C*-algebra by an action of a second-countable compact group with the tracial Rokhlin property with comparison that could be deduced from the properties of its original algebra: (1) stable rank one; (2) real rank zero; (3) $β$-comparison; (4) Winter's $n$-comparison; (5) $m$-almost divisible; (6) weakly ($m$,$n$)-divisible.

math.OA↗
arXiv2023

Non unital generalized tracially approximated C*-algebras

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.

math.OA↗
arXiv2023

A note on the weak tracial Rokhlin property for finite group actions on simple unital C*-algebras

In this paper, we show that one of the conditions in the definition of weak tracial Rokhlin property for finite group actions on simple unital C*-algebras can be replaced by a seemingly weaker condition, or a seemingly stronger condition. As a corollary, this condition is redundant whenever the C*-algebra is not purely infinite. We also give a sufficient condition for the weak tracial Rokhlin property for finite group actions on simple unital C*-algebras to imply the tracial Rokhlin property.

math.OA↗
arXiv2022

A remark on weak tracial approximation

In this paper, we point out that the definition of weak tracial approximation can be improved and strengthened. An example of weak tracial approximation is also provided.

math.OA↗
arXiv2022

Generalized Tracially Approximated C*-algebras

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}.

math.OA↗
arXiv2021

Some properties for certain generalized tracial approximated ${\rm C^*}$-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}.

math.OA↗
arXiv2021

Comparison properties of asymptotically tracially approximation C*-algebras

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).

math.OA↗
arXiv2019

Some Permanence for Large Subalgebra

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.

math.OA↗
arXiv2019

c-numerical range of operator products on B(H)

Let H be a complex Hilbert space of dimension no less than 2 and B(H) be the algebra of all bounded linear operators on H. We give the form of surjective maps on B(H) preserving c-numerical range of operator products when the maps satisfy preserving weak zero products. As a result, we obtain the characterization of surjective maps on Mn(C) preserving c-numerical range of operator products. The proof of the results depends on some propositions of operators in B(H), which are of different interest.

math.FA↗
arXiv2017

On majorization and range inclusion of operators on Hilbert $C^*$-modules

It is proved that for adjointable operators $A$ and $B$ between Hilbert $C^*$-modules, certain majorization conditions are always equivalent without any assumptions on $\overline{\mathcal{R}(A^*)}$, where $A^*$ denotes the adjoint operator of $A$ and $\overline{\mathcal{R}(A^*)}$ is the norm closure of the range of $A^*$. In the case that $\overline{{\mathcal R}(A^*)}$ is not orthogonally complemented, it is proved that there always exists an adjointable operator $B$ whose range is contained in that of $A$, whereas the associated equation $AX=B$ for adjointable operators is unsolvable.

math.OA↗
arXiv2005

$AF$ Embedding of Crossed Products of Certain Graph $C^*$-Algebras by Quasi-free Actions

We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph $C^*$-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an $AF$ algebra are discussed, and one sufficient condition is proved that if the row-finite directed graph is constructed by possibly attaching some 1-loops to a row-finite directed graph whose each weak connected component is a rooted (possibly infinite) directed tree, and the labelling map is almost proper, which is proved to be a reasonable generalization of the earlier case, then the crossed product can be embedded to an $AF$ algebra.

math.OA↗