Circle actions on UHF-absorbing $C^{\ast }$-algebras

Abstract

We study circle actions with the Rokhlin property, in relation to their restrictions to finite subgroups. We construct examples showing the following: the restriction of a circle action with the Rokhlin property (even on a real rank zero $C^{\ast }$-algebra), need not have the Rokhlin property; and even if every restriction of a given circle action has the Rokhlin property, the circle action itself need not have it. As a positive result, we show that the restriction of a circle action with the Rokhlin property to the subgroup ${\mathbb{Z}}_n$ has the Rokhlin property if the underlying algebra absorbs $M_{n^{\infty }}$. The condition on the algebra is also necessary in most cases of interest. Despite the fact that there are no circle actions with the Rokhlin property on UHF-algebras, we construct many such actions on certain UHF-absorbing simple AT-algebras. Additionally, we show that circle actions with the Rokhlin property on ${\mathcal{O}}_2$-absorbing $C^{\ast }$-algebras are generic, in a suitable sense.

24 pages

References in corpus (2)

• Freeness of actions of finite groups on C*-algebras
• The tracial Rokhlin property is generic

Cited by in corpus (3)

• KK-theory of circle actions with the Rokhlin property
• The radius of comparison of the crossed product by a tracially strictly approximately inner action
• Compact Group Actions with the Tracial Rokhlin Property