Homomorphisms of commutative Banach algebras and extensions to multiplier algebras with applications to Fourier algebras

E. Kaniuth¹ ; A. T. Lau² ; A. Ülger³

¹ Institut für Mathematik Universität Paderborn D-33095 Paderborn, Germany
² Department of Mathematical and Statistical Sciences University of Alberta Edmonton, Alberta, Canada T6G 2G1
³ Department of Mathematics Koc University 34450 Sariyer, Istanbul, Turkey

Studia Mathematica, Tome 183 (2007) no. 1, pp. 35-62

Résumé

Let $A$ and $B$ be semisimple commutative Banach algebras with bounded approximate identities. We investigate the problem of extending a homomorphism $\varphi : A \to B$ to a homomorphism of the multiplier algebras $M(A)$ and $M(B)$ of $A$ and $B$, respectively. Various sufficient conditions in terms of $B$ (or $B$ and $\varphi $) are given that allow the construction of such extensions. We exhibit a number of classes of Banach algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from $A$ into $A$ with closed range. Our results are applied to Fourier algebras of locally compact groups.

Détail

DOI : 10.4064/sm183-1-3

Keywords: semisimple commutative banach algebras bounded approximate identities investigate problem extending homomorphism varphi homomorphism multiplier algebras respectively various sufficient conditions terms varphi given allow construction extensions exhibit number classes banach algebras which these criteria apply addition prove polar decomposition homomorphisms closed range results applied fourier algebras locally compact groups