Spectral operation in locally convex algebras

Boukasmi, D. El; Kinani, A. El; Boukasmi, D. El; Kinani, A. El

doi:10.22199/issn.0717-6279-4548

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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.41 no.3 Antofagasta jun. 2022

http://dx.doi.org/10.22199/issn.0717-6279-4548

Artículos

Spectral operation in locally convex algebras

D. El Boukasmi¹

A. El Kinani²

^¹Université Mohammed V de Rabat, E. N. S de Rabat, B. P. 5118, 10105, Rabat, Maroc. e-mail: ddriss6@gmail.com

^²Université Mohammed V de Rabat, E. N. S de Rabat, B. P. 5118, 10105, Rabat, Maroc. e-mail: abdellah.elkinani@um5.ac.ma

Abstract

We show that if A is a spectrally bounded algebra, then all functions operate spectrally on A if and only if Sp_Ax is finite for every x ∈ A. We also prove that if A is a commutative Q-l.m.c.a, then all functions operate spectrally on A if and only if A/RadA is algebraic. Furthermore, if A is a semi-simple commutative Q-l.m.c.a. which is a Baire space, all functions operate spectrally on A if and only if it is isomorphic to C ⁿ for some n ∈ N. A structure result concerning semi-simple commutative complete m-convex algebras of countable dimension is also given.

Keywords and phrases: Spectrally bounded algebra; Q-algebra; l.m.c.a.; algebraic algebra; Baire space; Fourier-Gelfand transform; algebra of countable Hamel basis; operate function; function operate spectrally

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Received: October 30, 2020; Accepted: October 30, 2021

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