Generalized notion of amenability for a class of matrix algebras

TL;DR

This paper explores the conditions under which a class of upper triangular matrix algebras are amenable or pseudo-amenable, revealing that such properties hold only in trivial cases where the index set is a singleton.

Contribution

It establishes necessary and sufficient conditions for the amenability and pseudo-amenability of upper triangular matrix algebras based on the properties of the underlying Banach algebra and the index set.

Findings

UP(I,A) is amenable iff I is singleton and A is amenable.

UP(I,A) is pseudo-amenable iff I is singleton and A is pseudo-amenable.

Characterization of pseudo-contractibility and approximate biprojectivity for these algebras.

Abstract

We investigate the notions of amenability and its related homological notions for a class of $I\times I$-upper triangular matrix algebra, say $UP(I,A)$, where $A$ is a Banach algebra equipped with a non-zero character. We show that $UP(I,A)$ is pseudo-contractible (amenable) if and only if $I$ is singleton and $A$ is pseudo-contractible (amenable), respectively. We also study the notions of pseudo-amenability and approximate biprojectivity of $UP(I,A)$.