Hostname: page-component-669899f699-chc8l Total loading time: 0 Render date: 2025-04-26T19:40:36.237Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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

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)
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1. Alaminos, J., Brešar, M., Extremera, J. and Villena, A. R., Maps preserving zero products, Studia Math. 193 (2009), 131–159.CrossRefGoogle Scholar
2. Alaminos, J., Brešar, M., Extremera, J. and Villena, A. R., Characterizing Jordan maps on C*-algebras through zero products, Proc. Edinb. Math. Soc. 53 (2010), 543–555.Google Scholar
3. Alaminos, J., Brešar, M., Špenko, Š. and Villena, A. R., Derivations preserving quasinilpotent elements, Bull. Lond. Math. Soc. 46 (2014), 379–384.Google Scholar
4. Alaminos, J., Extremera, J. and Villena, A. R., Orthogonally additive polynomials on Fourier algebras, J. Math. Analysis Applic. 422(1) (2015), 72–83.Google Scholar
5. Benyamini, Y., Lassalle, S. and Llavona, J. G., Homogeneous orthogonally additive polynomials on Banach lattices, Bull. Lond. Math. Soc. 38 (2006), 459–469.Google Scholar
6. Bonsall, F. F. and Duncan, J., Complete normed algebras (Springer, 1973).Google Scholar
7. Boulabiar, K. and Buskes, G., Vector lattice powers: f-algebras and functional calculus, Commun. Alg. 34 (2006), 1435–1442.Google Scholar
8. Brešar, M. and Špenko, Š., Determining elements in Banach algebras through spectral properties, J. Math. Analysis Applic. 393 (2012), 144–150.CrossRefGoogle Scholar
9. Carando, D., Lassalle, S. and Zalduendo, I., Orthogonally additive polynomials over C(K) are measures—a short proof, Integ. Eqns Operat. Theory 56 (2006), 597–602.Google Scholar
10. Dales, H. G., Banach algebras and automatic continuity, London Mathematical Society Monographs Series, Oxford Science Publications, Volume 24 (Clarendon, Oxford, 2000).Google Scholar
11. Ibort, A., Linares, P. and Llavona, J. G., A representation theorem for orthogonally additive polynomials on Riesz spaces, Rev. Mat. Complut. 25 (2012), 21–30.CrossRefGoogle Scholar
12. Palmer, T. W., Banach algebras and the general theory of *-algebras, Volume 2: *-algebras, Encyclopedia of Mathematics and Its Applications, Volume 79 (Cambridge University Press, 2001).CrossRefGoogle Scholar
13. Pérez-García, D. and Villanueva, I., Orthogonally additive polynomials on spaces of continuous functions, J. Math. Analysis Applic. 306 (2005), 97–105.Google Scholar
14. Samei, E., Reflexivity and hyperreflexivity of bounded n-cocycles from group algebras, Proc. Am. Math. Soc. 139 (2011), 163–176.Google Scholar