Abstract view
A Short Note on the Continuous Rokhlin Property and the Universal Coefficient Theorem in $E$Theory


Published:20150303
Printed: Jun 2015
Gábor Szabó,
Westfälische WilhelmsUniversität, Fachbereich Mathematik, Einsteinstrasse 62, 48149 Münster, Germany
Abstract
Let $G$ be a metrizable compact group, $A$ a separable $\mathrm{C}^*$algebra
and $\alpha\colon G\to\operatorname{Aut}(A)$ a strongly continuous action.
Provided that $\alpha$ satisfies the continuous Rokhlin property,
we show that the property of satisfying the UCT in $E$theory
passes from $A$ to the crossed product $\mathrm{C}^*$algebra $A\rtimes_\alpha
G$ and the fixed point algebra $A^\alpha$. This extends a similar
result by Gardella for $KK$theory in the case of unital
$\mathrm{C}^*$algebras,
but with a shorter and less technical proof. For circle actions
on separable, unital $\mathrm{C}^*$algebras with the continuous Rokhlin
property, we establish a connection between the $E$theory equivalence
class of $A$ and that of its fixed point algebra $A^\alpha$.