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Maximal abelian subalgebras of Banach algebras

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 .