
doi: 10.1112/blms.12551
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 \(C\).
uniform algebra, function algebra, maximal abelian subalgebra, Ideals and subalgebras, commutative Banach algebra, 510
uniform algebra, function algebra, maximal abelian subalgebra, Ideals and subalgebras, commutative Banach algebra, 510
| 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). | 0 | |
| 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. | Average | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Average | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Average |
