Unique ergodicity of free shifts and some other automorphisms of C*-algebras

TL;DR

This paper introduces a new notion of unique ergodicity relative to fixed-point subalgebras for automorphisms of unital C*-algebras and proves it for free shifts and certain group automorphisms, with a generalization of Haagerup's inequality.

Contribution

It defines a relative ergodicity concept and proves it for free shifts and specific group automorphisms, extending previous results with a new inequality.

Findings

Free shift on reduced amalgamated free product C*-algebras is uniquely ergodic relative to its fixed-point subalgebra.

Automorphisms of reduced group C*-algebras from certain group automorphisms are uniquely ergodic.

A generalized Haagerup's inequality provides bounds on norms in reduced amalgamated free product C*-algebras.

Abstract

A notion of unique ergodicity relative to the fixed-point subalgebra is defined for automorphisms of unital C*-algebras. It is proved that the free shift on any reduced amalgamated free product C*-algebra is uniquely ergodic relative to its fixed-point subalgebra, as are autormorphisms of reduced group C*-algebras arising from certain automorphisms of groups. A generalization of Haagerup's inequality, yielding bounds on the norms of certain elements in reduced amalgamated free product C*-algebras, is proved.