Homological and cohomological properties of Banach algebras and their second duals (2210.16596v1)

Abstract: In this paper, we investigate homological properties of Banach algebras. We show that retractions Banach algebras preserve biprojectivity, contractibility and biflatness. We also prove that contractibility of second dual of a Banach algebra implies contractibility of the Banach algebra. For a Banach algebra $A$ with $\Delta(A)\neq\emptyset$, let $\frak{F}(X, A)$ be one of the Banach algebras $C_b(X, A)$, $C_0(X, A)$, $\hbox{Lip}\alpha(X, A)$ or $\hbox{lip}\alpha(X, A)$. In the following, we study homological properties of Banach algebra $\frak{F}(X, A)$, especially contractibility of it. We prove that contractibility of $\frak{F}(X, A)$ is equivalent to finiteness of $X$ and contractibility of $A$. In the case where, $A$ is commutative, we show that $\frak{F}(X, A)$ is contractible if and only if $A$ is a $C^*-$algebra and both $X$ and $\Delta(A)$ are finite. In particular, $\hbox{lip}_\alpha^0(X, A)$ is contractible if and only if $X$ is finite. We also investigate contractibility of $L^1(G, A)$ and establish $L^1(G, A)$ is contractible if and only if $G$ finite and $A$ is contractible. Finally, we show that biprojectivity of the Beurling algebra $L^1(G, \omega)$ is equivalent to compactness of $G$, however, biprojectivity of the Banach algebras $L^1(G, \omega)^{**}$ is equivalent to finiteness of $G$. This result holds for the Banach algebra $M(G, \omega)^{**}$ instead of $L^1(G, \omega)^{**}$.