Abstract view
The JiangSu absorption for inclusions of unital C*algebras


Published:20171206
Printed: Apr 2018
Hiroyuki Osaka,
Department of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 5258577 Japan
Tamotsu Teruya,
Faculty of Education, Gunma University, 42 Aramakimachi, Maebashi City, Gunma, 3718510, Japan
Abstract
We introduce the tracial Rokhlin property for a conditional expectation
for an inclusion
of unital C*algebras $P \subset A$ with index finite, and show
that an action $\alpha$
from a finite group $G$ on a simple unital C*algebra $A$ has
the tracial Rokhlin property
in the sense of N. C. Phillips
if and only if the canonical conditional expectation $E\colon
A \rightarrow A^G$ has the tracial
Rokhlin property.
Let $\mathcal{C}$ be a class of infinite dimensional stably
finite separable unital C*algebras
which is closed under the following conditions:
(1)
If $A \in {\mathcal C}$ and $B \cong A$, then $B \in \mathcal{C}$.
(2)
If $A \in \mathcal{C}$ and $n \in \mathbb{N}$, then $M_n(A) \in \mathcal{C}$.
(3)
If $A \in \mathcal{C}$ and $p \in A$ is a nonzero projection,
then $pAp \in \mathcal{C}$.
Suppose that any C*algebra in $\mathcal{C}$ is weakly semiprojective.
We prove that if $A$ is a local tracial $\mathcal{C}$algebra
in the sense of Fan and Fang and a conditional expectation
$E\colon A \rightarrow P$ is of indexfinite type with the tracial
Rokhlin property, then $P$ is a unital
local tracial $\mathcal{C}$algebra.
The main result is that if $A$ is simple, separable, unital
nuclear, JiangSu absorbing
and $E\colon A \rightarrow P$ has the tracial Rokhlin property,
then $P$ is JiangSu absorbing.
As an application, when an action $\alpha$
from a finite group $G$ on a simple unital C*algebra $A$ has
the tracial Rokhlin property,
then for any subgroup $H$ of $G$ the fixed point algebra $A^H$
and the crossed product algebra
$A \rtimes_{\alpha_{H}} H$ is JiangSu absorbing.
We also show that the strict comparison property for a Cuntz
semigroup $W(A)$
is hereditary to $W(P)$ if $A$ is simple, separable, exact,
unital,
and $E\colon A \rightarrow P$ has the tracial Rokhlin property.