Singly generated operator algebras satisfying weakened versions of amenability

TL;DR

This paper constructs specific singly generated operator algebras that challenge traditional notions of amenability, showing that certain weakened properties do not imply full amenability.

Contribution

It provides explicit examples of singly generated subalgebras with weakened amenability properties, expanding understanding of operator algebra structures.

Findings

Constructed a non-amenable, boundedly approximately contractible subalgebra of compact operators.

Identified singly generated, biflat subalgebras of finite Type I von Neumann algebras that are not amenable.

Demonstrated that certain extension properties do not imply amenability.

Abstract

We construct a singly generated subalgebra of ${\mathcal K}({\mathcal H})$ which is non-amenable, yet is boundedly approximately contractible. The example embeds into a homogeneous von Neumann algebra. We also observe that there are singly generated, biflat subalgebras of finite Type I von Neumann algebras, which are not amenable (and hence are not isomorphic to C*-algebras). Such an example can be used to show that a certain extension property for commutative operator algebras, which is shown in arXiv:1012.4259 to follow from amenability, does not necessarily imply amenability.