Stable and real rank for crossed products by automorphisms with the tracial Rokhlin property

By Hiroyuki Osaka, N. Christopher Phillips. 2004

60 citations recorded by the source corpus.

We introduce the tracial Rokhlin property for automorphisms of stably finite simple unital C*-algebras containing enough projections. This property is formally weaker than the various Rokhlin properties considered by Herman and Ocneanu, Kishimoto, and Izumi. Our main results are as follows. Consider a stably finite simple unital C*-algebra, and an automorphism of it which has the tracial Rokhlin property. Suppose the algebra has real rank zero and stable rank one, and suppose that the order on projections over the algebra is determined by traces. Then the crossed product by the automorphism also has these three properties. We also present examples of C*-algebras and automorphisms which satisfy the above assumptions, but such that crossed product algebras do not have tracial rank zero.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/297c829b518ab5223d81fe1359446943181ce560

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Mathematical Physics

Citations in the corpus

The tracial Rokhlin property for actions of finite groups on C*-algebras (2006, methodology): “…there exists ε0 > 0 such that whenever (xg)g∈G is a collection of elements of a unital C*-algebra B such that for all g, h ∈ G we have ‖xgx ∗ g − 1‖ 10ε0, ‖x ∗ gxg − 1‖ 10ε0, and ‖xgxh − xgh‖ 10ε0, then there exists a unital homomorphism ϕ : C∗(G) → B such that ‖ϕ(ug)−xg‖ < 1 13ε…”
The Rohlin property for automorphisms of the Jiang-Su algebra (2009, background): “Similar definitions for finite actions, which is called projection free tracial Rohlin property, has been defined in [1], [25].”
Finite cyclic group actions with the tracial Rokhlin property (2006, background): “This is true for actions of Z on stably finite simple unital C*-algebras, by Theorems 3.5, 4.5, and 5.3 of [ 20 ].”
Rokhlin Dimension and C*-Dynamics (2012, background): “We note that other generalizations of the Rokhlin property have been considered which are based on the idea of leaving out a remainder which is small in the tracial sense (see [19, 25, 1, 6, 22]).”
Freeness of actions of finite groups on C*-algebras (2009, methodology): “The analog for actions of Z is in [49].”
Tracially Z-absorbing C*-algebras (2012, background): “The Rokhlin property has been generalized to a tracial version in [15, 17], although this generalization still requires the existence of projections.”