Geometry of Banach algebra $\mA$ and the bidual of $L^1(G,\mA)$

TL;DR

This paper investigates the structure of the bidual of generalized group algebras $L^1(G,\mathcal{A})$ with Arens products, extending known results about topological centers to Banach algebra valued cases and exploring properties like the pseudo-center.

Contribution

It extends the characterization of topological centers from scalar to Banach algebra valued cases and analyzes the pseudo-center for non-reflexive algebras using vector measures.

Findings

The left topological center of $(L^1(G)\hat\otimes\mathcal{A})^{**}$ is a Banach $L^1(G)$-module for abelian $G$.

The topological center of $L^1(G)^{**}$ is $L^1(G)$, extended to Banach algebra valued case.

Partial characterization of elements in the pseudo-center using Cohen's factorization theorem.

Abstract

This article is intended towards the study of the bidual of generalized group algebra $L^1(G,\mA)$ equipped with two Arens product, where $G$ is any locally compact group and $\mA$ is a Banach algebra. We show that the left topological center of $(L^1(G)\hat\otimes\mA)^{**}$ is a Banach $L^1(G)$-module if $G$ is abelian. Further it also holds permanance property with respect to the unitization of $\mA$. We then use this fact to extend the remarkable result of A.M Lau and V. Losert\cite{Lau-losert}, about the topological center of $L^1(G)^{**}$ being just $L^1(G)$, to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of $L^1(G,\mA)$ for non-reflexive Banach algebras $\mA$ and give a partial characterization for elements of pseudo-center using the Cohen's factorization theorem. In the running we also observe few consequences when $\mA$ holds the Radon-Nikodym property and weak sequential completeness.