Maps preserving zero products

J. Alaminos, et al. "Maps preserving zero products." Studia Mathematica 193.2 (2009): 131-159. <http://eudml.org/doc/284931>.

@article{J2009,
abstract = {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.},
author = {J. Alaminos, M. Brešar, J. Extremera, A. R. Villena},
journal = {Studia Mathematica},
keywords = {group algebra; -algebra; homomorphism; weighted homomorphism; derivation; generalized derivation; separating map; disjointness preserving map; zero product preserving map; doubly power-bounded element},
language = {eng},
number = {2},
pages = {131-159},
title = {Maps preserving zero products},
url = {http://eudml.org/doc/284931},
volume = {193},
year = {2009},
}

TY - JOUR
AU - J. Alaminos
AU - M. Brešar
AU - J. Extremera
AU - A. R. Villena
TI - Maps preserving zero products
JO - Studia Mathematica
PY - 2009
VL - 193
IS - 2
SP - 131
EP - 159
AB - 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.
LA - eng
KW - group algebra; -algebra; homomorphism; weighted homomorphism; derivation; generalized derivation; separating map; disjointness preserving map; zero product preserving map; doubly power-bounded element
UR - http://eudml.org/doc/284931
ER -