Maximal abelian subalgebras of Banach algebras

Abstract

Let A be a commutative, unital Banach algebra. We consider the number of different non-commutative, unital Banach algebras C such that A is a maximal abelian subalgebra in C. For example, we shall prove that, in the case where A is an infinite-dimensional, unital Banach function algebra, A is a maximal abelian subalgebra in infinitely-many closed subalgebras of B ( A ) such that no two distinct subalgebras are isomorphic; the same result holds for certain examples A that are local algebras. We shall also give examples of uniform algebras of the form C ( K ) , where K is a compact space, with the property that there exists a family of arbitrarily large cardinality of pairwise non-isomorphic unital Banach algebras C such that each C contains B ( C ( K ) ) as a closed subalgebra and is such that C ( K ) is a maximal abelian subalgebra in C.