Trace Asymptotics for C*-Algebras from Smale Spaces

TL;DR

This paper studies the asymptotic behavior of traces on C*-algebras derived from hyperbolic dynamical systems called Smale spaces, revealing how these traces can be obtained via limits in mixing cases.

Contribution

It introduces a method to compute traces on C*-algebras associated with Smale spaces using a limiting process, expanding understanding of their structure.

Findings

Traces on these C*-algebras can be obtained through a limiting process.

In mixing Smale spaces, the usual operator trace can be recovered as a limit.

The algebraic structure exhibits compact products and a unifying unitary operator.

Abstract

We consider C*-algebras associated with stable and unstable equivalence in hyperbolic dynamical systems known as Smale spaces. These systems include shifts of finite type, in which case these C*-algebras are both AF-algebras. These algebras have fundamental representations on a single Hilbert space (subject to a choice of periodic points) which have a number of special properties. In particular, the product between any element of the first algebra with one from the second is compact. In addition, there is a single unitary operator which implements actions on both. Here, under the hypothesis that the system is mixing, we show that the (semi-finite) traces on these algebras may be obtained through a limiting process and the usual operator trace.