Maps preserving zero products

• Autorzy: J. Alaminos , M. Brešar , J. Extremera , A. R. Villena
• Języki publikacji: EN

Abstrakty

EN

A linear map T from a Banach algebra A into another B preserves zero products if T(a)T(b) = 0 whenever a,b ∈ A are such that ab = 0. This paper is mainly concerned with the question of whether every continuous linear surjective map T: A → 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 ϕ: A × A → X (for some Banach space X) with the property that ϕ(a,b) = 0 whenever a,b ∈ A are such that ab = 0. We prove that such a map necessarily satisfies ϕ(aμ,b) = ϕ(a,μ b) for all a,b ∈ A and for all μ from the closure with respect to the strong operator topology of the subalgebra of ℳ(A) (the multiplier algebra of A) generated by doubly power-bounded elements of ℳ(A). This method is also shown to be useful for characterizing derivations through the zero products.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2009
• Tom: 193
• Numer: 2
• Strony: 131-159

Daty

wydano

2009

Twórcy

autor

J. Alaminos

• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

autor

M. Brešar

• Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana, Slovenia
• Faculty of Natural Sciences and Mathematics, University of Maribor, 2000 Maribor, Slovenia

autor

J. Extremera

• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

autor

A. R. Villena

• Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

Identyfikatory

DOI

10.4064/sm193-2-3