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) $\beta$-comparison; (4) Winter's $n$-comparison; (5) $m$-almost divisible; (6) weakly ($m$,$n$)-divisible.Comment: final version, 24 pages. This version was accepted in Rocky Mountain Journal of Mathematic

The inheritance tracial topological rank zero for centrally large subalgebra

Let $A$ be an infinite-dimensional unital simple C*-algebra such that A has $\mu$-$oz{\rm LLP}$ property for any $\mu\in(0,1)$. 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.Comment: arXiv admin note: substantial text overlap with arXiv:2309.08900, arXiv:2203.05700, arXiv:2310.12548, arXiv:2310.04718, arXiv:2206.0503

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.Comment: 14page