Some homological properties on generalized amalgamated Banach algebras

Abstract

In the present paper, first we characterize the multiplier algebra, (maximal) ideals, minimal idempotents, spectrum and radical of the generalized amalgamated Banach algebra $A\boxtimes_\Theta X$ in terms of $A$ and $X$ and the bilinear mapping $\Theta$ from $X\times X$ into $A$. Then, we show that there are a strong relationship between some of homological properties of $A\boxtimes_\Theta X$, such as Connes-amenability, flatness and projectivity, $\phi$-biprojectivity, and the Banach algebras $A$ and $X$ and the mapping $\Theta$. The results of this paper extend several results in the literature.