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Bulletin of the London Mathematical Society
Article . 2021 . Peer-reviewed
License: Wiley Online Library User Agreement
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image/svg+xml Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao Closed Access logo, derived from PLoS Open Access logo. This version with transparent background. http://commons.wikimedia.org/wiki/File:Closed_Access_logo_transparent.svg Jakob Voss, based on art designer at PLoS, modified by Wikipedia users Nina and Beao
zbMATH Open
Article . 2021
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Maximal abelian subalgebras of Banach algebras

Authors: Dales, H.G.; Pham, H.L.; Żelazko, W.;

Maximal abelian subalgebras of Banach algebras

Abstract

The present paper begins by showing that if \(A\) is a commutative unital Banach algebra whose character space has cardinality greater than one, then there are families of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras that contain \(A\) as a maximal abelian subalgebra. Motivated by this property, the authors address the following questions for an infinite-dimensional, commutative, unital Banach algebra \(A\): (i) How many pairwise non-isomorphic, closed, non-commutative, unital subalgebras \(C\) of \(\mathcal{B}(A)\) (the Banach algebra of all bounded linear operators on \(A\)) are such that \(A\) is a maximal abelian subalgebra of \(C\)? (ii) How many pairwise non-isomorphic, non-commutative, unital Banach algebras \(C\) are there that contain \(\mathcal{B}(A)\) as a closed, unital subalgebra and are such that \(A\) is a maximal abelian subalgebra of \(C\)? Concerning the first question, the authors show that in the case where \(A\) is an infinite-dimensional function algebra, \(A\) is a maximal abelian subalgebra of infinitely-many pairwise non-isomorphic closed subalgebras of \(\mathcal{B}(A)\). The authors give a significant answer to the second question by showing that there are compact spaces \(K\) and a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras \(C\) such that each \(C\) contains \(\mathcal{B}(C(K))\) as a closed subalgebra and such that \(C(K)\) is a maximal abelian subalgebra in each \(C\).

Country
United Kingdom
Keywords

uniform algebra, function algebra, maximal abelian subalgebra, Ideals and subalgebras, commutative Banach algebra, 510

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
0
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