Homological propeties of Bimeasure algebras and their BSE properties

TL;DR

This paper explores the homological properties of Bimeasure algebras on locally compact groups, establishing conditions for amenability, biprojectivity, and BSE properties based on the discreteness and finiteness of the groups.

Contribution

It provides new characterizations of Bimeasure algebras' homological properties, linking amenability, biprojectivity, and BSE properties to group discreteness and finiteness.

Findings

BM(G, H) is amenable iff G and H are discrete.

Biprojectivity of BM(G, H) is equivalent to G and H being finite.

BM(G, H) is a BSE algebra iff G and H are discrete groups.

Abstract

Let $G$ and $H$ be locally compact groups. $BM(G, H)$ denoted the Banach algebra of bounded bilinear forms on $C_{0}(G)\times C_{0}(H)$.In this paper, the homological properties of Bimeasure algebras are investigated. It is found and approved that the Bimeasure algebras $BM(G, H)$ is amenable if and only if $G$ and $H$ are discrete. The correlation between the weak amenability of $BM(G, H)$ and $M(G\times H)$ is assessed. It is found and approved that the biprojectivity of the bimeasure algebra $BM(G, H)$ is equivalent to the finiteness of $G$ and $H$. Furthermore, we show that the bimeasure group algebra $BM_{a}(G, H)$ is a BSE algebra. It will be concluded that $BM(G, H)$ is a BSE- algebra if and only if $G$ and $H$ are discrete groups.