Amenable-like and Connes Amenable-like Propertiesof Certain Matrix Algebras

Abstract

In this paper, we study some amenability like properties for certain matrix algebras called anti-symmetric matrix algebras, say $AS_I(\Bbb C)$, where the index set $I$ is a totally ordered set which posses a smallest element. We show that anti-symmetric matrix algebra is biprojective and weakly amenable. We prove that $AS_I(\Bbb C)$ is pseudo-contractible if and only if $I$ is singleton. In the second part of the paper we investigate some amenability like properties for a class of $\ell^{2}$-Munn algebras. As a dual Banach algebra, we study some Connes homological properties of these algebras like Connes biprojectivity and approximate Connes amenability.