Some homological properties of T-Lau product algebra

TL;DR

This paper studies various homological properties of the T-Lau product algebra formed from two Banach algebras, revealing how these properties behave and providing characterizations and examples.

Contribution

It introduces the analysis of approximate amenability, pseudo amenability, and related properties for T-Lau product algebras, including stability and characterization results.

Findings

Approximate amenability of A and B is not necessarily preserved in A *_T B.

Characterization of the double centralizer algebra of A *_T B.

An application demonstrating the use of the algebra's properties.

Abstract

Let T be a homomorphism from a Banach algebra B to a Banach algebra A.The Cartesian product space A * B with T-Lau multiplication and l^1-norm becomes a new Banach algebra A *_T B. We investigate the notions such as approximate amenability, pseudo amenability, phi-pseudo amenability,phi-biflatness and phi-biprojectivity for Banach algebra A *_T B. We also present an example to show that approximate amenability of A and B is not stable for A *_T B. Finally we characterize the double centralizer algebra of A *_T B and present an application of this characterization.