George A. Elliott, Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu

Qingzhai Fan , Department of Mathematics, Shanghai Maritime University, Shangha, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang, Department of Mathematics, Tongji University, Shanghai, China 200092; e-mail: xfang@tongji.edu.cn

Abstract/Résumé:

In this paper, we introduce some classes of generalized tracial approximation C*-algebras. Consider the class of unital C*-algebras which are tracially 𝒵-absorbing (or have tracial nuclear dimension at most n, or have the property SP, or are m-almost divisible). Then A is tracially 𝒵-absorbing (respectively, has tracial nuclear dimension at most n, has the property SP, is weakly (n, m)-almost divisible) for any simple unital C*-algebra A in the corresponding class of generalized tracial approximation C*-algebras. As an application, let A be an infinite-dimensional unital simple C*-algebra, and let B be a centrally large subalgebra of A. If B is tracially 𝒵-absorbing, then A is tracially 𝒵-absorbing. This result was obtained by Archey, Buck, and Phillips in Archey et al. (2018).

On introduit la notion d’approximation traciale généralisée d’une C*-algèbre par des C*-algèbres dans une class donnée. Cette notion généralise la notion de Lin d’approximation triviale simple, et aussi la notion d’Archey et de Phillips de centralement grande sousalgèbre, deux notions qui se sont démontrées très importantes.

Keywords: Cuntz Semigroup, C∗-algebras, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Generalized Tracially Approximated C*-algebras

George A. Elliott, FRSC,Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 2E4; e-mail: elliott@math.toronto.edu,

Qingzhai Fan,Department of Mathematics, Shanghai Maritime University, Shanghai, China 201306; e-mail: fanqingzhai@fudan.edu.cn,qzfan@shmtu.edu.cn

Xiaochun Fang,Department of Mathematics, Tongji University, Shangha, China 200092; e-mail: xfang@mail.tongji.edu.cn

Abstract/Résumé:

We show that the following properties of the \({\rm C^*}\)-algebras in a class \(\Omega\) are inherited by simple unital \({\rm C^*}\)-algebras in the class \({\rm TA}\Omega\): \((1)\) \(\beta\)-comparison (\(1\leq \beta < \infty\)), \((2)\) \(n\)-comparison, \((3)\) trace \(\mathcal{Z}\)– absorption, \((4)\) \(m\)-almost divisibility, \((5)\) \((n,m) ~(m\neq 0)\) comparison, and \((6)\) tracial approximate divisibility. As an application, every unital simple \({\rm C^*}\)-algebra with tracial topological rank at most \(k\) has the property of \(k\)-comparison. Also as an application, let \(A\) be an infinite-dimensional simple unital \({\rm C^*}\)-algebra such that \(A\) has one of the above-listed properties. Suppose that \(\alpha: G\to {\rm Aut}(A)\) is an action of a finite group \(G\) on \(A\) which has the tracial Rokhlin property. Then the crossed product \({\rm C^*}\)-algebra \({\rm C^*}( G, A,\alpha)\) also has the property under consideration.

On considère plusieurs propriétés d’une C*-algèbre simple à élément unité qui sont héritées par approximation traciale. Comme application on démontre que ces propriétés sont aussi héritées par la C*-algèbre produit croisé associée à une action d’un groupe fini qui possède la propriété de Rokhlin traciale.

Keywords: C*-algebra, Cuntz Semigroup, tracial approximation

AMS Subject Classification: General theory of $C^*$-algebras, Classifications of $C^*$-algebras; factors, K-theory and operator algebras -including cyclic theory 46L05, 46L35, 46L80

PDF(click to download): Certain Properties of Tracial Approximation ${C^*}$-Algebras