Homological and cohomological properties of Banach algebras and their second duals

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 $\Delta (A)\ne \varnothing$, let $\mathfrak{F}(X,A)$ be one of the Banach algebras $C_b(X,A)$, $C_0(X,A)$, ${Lip}_{\alpha }(X,A)$ or ${lip}_{\alpha }(X,A)$. In the following, we study homological properties of Banach algebra $\mathfrak{F}(X,A)$, especially contractibility of it. We prove that contractibility of $\mathfrak{F}(X,A)$ is equivalent to finiteness of $X$ and contractibility of $A$. In the case where, $A$ is commutative, we show that $\mathfrak{F}(X,A)$ is contractible if and only if $A$ is a $C^{\ast }-$algebra and both $X$ and $\Delta (A)$ are finite. In particular, ${lip}_{\alpha }^0(X,A)$ 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 $L^1(G,\omega {)}^{\ast \ast }$ is equivalent to finiteness of $G$. This result holds for the Banach algebra $M(G,\omega {)}^{\ast \ast }$ instead of $L^1(G,\omega {)}^{\ast \ast }$.