Weak tolpological centers and cohomological properties

TL;DR

This paper introduces the concept of weak topological centers in Banach algebra modules, explores their properties and differences from traditional centers, and applies these ideas to cohomological and amenability properties of Banach algebras.

Contribution

It defines weak topological centers for Banach modules, investigates their properties, and connects these concepts to cohomology and amenability in Banach algebras.

Findings

Weak topological centers differ from classical topological centers.

For a compact group, specific equalities involving $L^1(G)$ and measure algebras are established.

Applications to weak amenability and cohomological properties are demonstrated.

Abstract

Let $B$ be a Banach $A-bimodule$. We introduce the weak topological centers of left module action and we show it by $\tilde{{Z}}^\ell_{B^{**}}(A^{**})$. For a compact group, we show that $L^1(G)=\tilde{Z}_{M(G)^{**}}^\ell(L^1(G)^{**})$ and on the other hand we have $\tilde{Z}_1^\ell{(c_0^{**})}\neq c_0^{**}$. Thus the weak topological centers are different with topological centers of left or right module actions. In this manuscript, we investigate the relationships between two concepts with some conclusions in Banach algebras. We also have some application of this new concept and topological centers of module actions in the cohomological properties of Banach algebras, spacial, in the weak amenability and $n$-weak amenability of Banach algebras.