Homological propeties of Bimeasure algebras and their BSE properties

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 $B{M}_a(G,H)$ 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.