Triviality of the generalized Lau product associated to a Banach algebra homomorphism

TL;DR

This paper proves that the generalized Lau product of Banach algebras is essentially the same as their direct product, clarifying misconceptions and simplifying the understanding of this algebraic construction.

Contribution

It shows the generalized Lau product is isomorphic to the direct product of Banach algebras, correcting previous misconceptions about its structure.

Findings

Generalized Lau product is isomorphic to the direct product

Clarifies the relationship between Lau product and Monfared's construction

Simplifies the conceptual understanding of the Lau product

Abstract

Several papers have, as their raison d'etre, the exploration of the \emph{generalized Lau product} associated to a homomorphism $T:B\to A$ of Banach algebras. In this short note, we demonstrate that the generalized Lau product is isomorphic as a Banach algebra to the usual direct product $A\oplus B$. We also correct some misleading claims made about the relationship between this generalized Lau product, and an older construction of Monfared (Studia Mathematica, 2007).