Maps preserving zero products

J. Alaminos¹ ; M. Brešar² ; J. Extremera¹ ; A. R. Villena¹

¹Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
²Faculty of Mathematics and Physics University of Ljubljana 1000 Ljubljana, Slovenia and Faculty of Natural Sciences and Mathematics University of Maribor 2000 Maribor, Slovenia

Studia Mathematica, Tome 193 (2009) no. 2, pp. 131-159

Résumé

A linear map $T$ from a Banach algebra $A$ into another $B$ preserves zero products if $T(a)T(b)=0$ whenever $a,b\in A$ are such that $ab=0$. This paper is mainly concerned with the question of whether every continuous linear surjective map $T\colon A\rightarrow B$ that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras.Our method involves continuous bilinear maps $\phi\colon A\times A\rightarrow X$ (for some Banach space $X$) with the property that $\phi(a,b)=0$ whenever $a,b\in A$ are such that $ab=0$. We prove that such a map necessarily satisfies $\phi(a\mu,b)=\phi(a,\mu b)$ for all $a,b\in A$ and for all $\mu$ from the closure with respect to the strong operator topology of the subalgebra of $\mathcal{M}(A)$ (the multiplier algebra of $A$) generated by doubly power-bounded elements of $\mathcal{M}(A)$. This method is also shown to be useful for characterizing derivations through the zero products.

Détail

DOI : 10.4064/sm193-2-3

Keywords: linear map banach algebra another preserves zero products b whenever paper mainly concerned question whether every continuous linear surjective map colon rightarrow preserves zero products weighted homomorphism indeed large class banach algebras which includes group algebras method involves continuous bilinear maps phi colon times rightarrow banach space property phi whenever prove map necessarily satisfies phi phi for closure respect strong operator topology subalgebra mathcal multiplier algebra generated doubly power bounded elements mathcal method shown useful characterizing derivations through zero products