We introduce the tracial Rokhlin property for a conditional expectation for an inclusion of unital
${{\text{C}}^{*}}$
-algebras
$P\,\subset \,A$
with index finite, and show that an action
$\alpha$
from a finite group
$G$
on a simple unital
${{\text{C}}^{*}}$
- algebra
$A$
has the tracial Rokhlin property in the sense of N. C. Phillips if and only if the canonical conditional expectation
$E:\,A\,\to \,{{A}^{G}}\,$
has the tracial Rokhlin property. Let
$\mathcal{C}$
be a class of infinite dimensional stably finite separable unital
${{\text{C}}^{*}}$
-algebras that 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
${{M}_{n}}\left( A \right)\,\in \,\mathcal{C}$
.
(3) If
$A\,\in \,\mathcal{C}$
and
$p\,\in \,A$
is a nonzero projection, then
$pAp\,\in \,\mathcal{C}$
.
Suppose that any
${{\text{C}}^{*}}$
-algebra in
$\mathcal{C}$
is weakly semiprojective. We prove that if
$A$
is a local tracial
${{\text{C}}^{*}}$
-algebra in the sense of Fan and Fang and a conditional expectation
$E:\,A\,\to \,P$
is of index-finite 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, Jiang–Su absorbing and
$E:\,A\,\to \,P$
has the tracial Rokhlin property, then
$P$
is Jiang–Su absorbing. As an application, when an action α from a finite group
$G$
on a simple unital
${{\text{C}}^{*}}$
-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 Jiang–Su absorbing. We also show that the strict comparison property for a Cuntz semigroup
$W\left( A \right)$
is hereditary to
$W\left( P \right)$
if
$A$
is simple, separable, exact, unital, and
$E:\,A\,\to \,P$
has the tracial Rokhlin property.