Generalized notion of amenability for a class of matrix algebras

Abstract

We investigate the notions of amenability and its related homological notions for a class of I × I-upper triangular matrix algebra, say U P (I, A), where A is a Banach algebra equipped with a nonzero character. We show that U P (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 U P (I, A).Recently approximate versions of amenability and homological properties of Banach algebras have been under more observations. In [24] Zhang introduced the notion of approximately biprojective Banach algebras, that is, A is approximately biprojective if there exists a net of A-bimodule morphism ρAuthor with A. Pourabbas investigated approximate biprojectivity of 2 × 2 upper triangular Banach algebra which is a matrix algebra, also we characterized approximate biprojectivity of Segal algebras and weighted group algebras. We show that a Segal algebra S(G) is approximately biprojective if and only if