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Orthogonally Additive Polynomials and Orthosymmetric Maps in Banach Algebras with Properties 𝔸 and 𝔹

Part of: Abstract harmonic analysis Topological algebras, normed rings and algebras, Banach algebras Commutative Banach algebras and commutative topological algebras Special classes of linear operators

Published online by Cambridge University Press:  15 December 2015

J. Alaminos ,
M. Brešar ,
Š. Špenko  and
A. R. Villena
J. Alaminos
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; avillena@ugr.es)
M. Brešar
Affiliation:
Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia (matej.bresar@fmf.uni-lj.si) Faculty of Natural Sciences and Mathematics, University of Maribor, Korosca 160, 2000 Maribor, Slovenia (bresar@uni-mb.si)
Š. Špenko
Affiliation:
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia (spela.spenko@imfm.si)
A. R. Villena
Affiliation:
Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain (alaminos@ugr.es; avillena@ugr.es)
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Abstract

This paper considers Banach algebras with properties 𝔸 or 𝔹, introduced recently by Alaminos et al. The class of Banach algebras satisfying either of these two properties is quite large; in particular, it includes C *-algebras and group algebras on locally compact groups. Our first main result states that a continuous orthogonally additive n-homogeneous polynomial on a commutative Banach algebra with property 𝔸 and having a bounded approximate identity is of a standard form. The other main results describe Banach algebras A with property 𝔹 and having a bounded approximate identity that admit non-zero continuous symmetric orthosymmetric n-linear maps from An into ℂ.

Keywords

Banach algebrazero productproperty 𝔸property 𝔹orthogonally additive polynomialorthosymmetric map

MSC classification

Secondary: 43A20: $L^1$-algebras on groups, semigroups, etc. 46H05: General theory of topological algebras 46J05: General theory of commutative topological algebras 47B48: Operators on Banach algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 3 , August 2016 , pp. 559 - 568
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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