Amenability for unitary groups of simple monotracial C*-algebras

TL;DR

This paper investigates the amenability properties of unitary groups in certain classes of C*-algebras, proving right amenability and skew-amenability under specific conditions, thus addressing open conjectures and questions.

Contribution

It establishes new amenability results for unitary groups of simple monotracial C*-algebras and nuclear C*-algebras, partially resolving conjectures and questions in the field.

Findings

Isometry semigroup of unital properly infinite nuclear C*-algebra is right amenable

Unitary group of unital simple monotracial C*-algebra with hyperfinite tracial GNS is skew-amenable

Addresses conjecture of Alekseev, Schmidt, and Thom and question of Pestov

Abstract

We prove the following two results. First, the isometry semigroup of a unital properly infinite nuclear C*-algebra is right amenable. Second, the unitary group of a unital simple monotracial C*-algebra whose tracial GNS representation is hyperfinite is skew-amenable in the weak topology. This answers in part a conjecture of Alekseev, Schmidt, and Thom and a question of Pestov.