A characterization of multiplicative linear functionals in complex Banach algebras

A characterization of characters in complex m-pseudoconvex algebras

“…Equivalently: a linear functional f on A is multiplicative (a character) if and only if the value f (x) belongs to the spectrum σ(x) for each element x in A. This result was extended in [8] to the non-commutative case by showing that if f is a linear fuctional on a unital algebra A, which is multiplicative on each commutative subalgebra of A, then it is a character on A. It was already mentioned in [2] that the above result holds true for (commutative) complete locally bounded algebras and for (commutative) complete locally multiplicatively convex (m-convex) algebras, the latter under an additional assumption of continuity of f or closedness of M (the non-commutative versions of these results follow immediately from [8]).…”

“…This result was extended in [8] to the non-commutative case by showing that if f is a linear fuctional on a unital algebra A, which is multiplicative on each commutative subalgebra of A, then it is a character on A. It was already mentioned in [2] that the above result holds true for (commutative) complete locally bounded algebras and for (commutative) complete locally multiplicatively convex (m-convex) algebras, the latter under an additional assumption of continuity of f or closedness of M (the non-commutative versions of these results follow immediately from [8]). In this paper we give a common generalization of these two results, by showing that the above-mentioned holds true for complete complex unital locally multiplicatively pseudoconvex algebras.…”

“…so that |f (x)|/C ≤ 1 for all x in B. Consequently, (8) |f ( B o is an idempotent set. But the convex envelope of an idempotent set is again idempotent (see [3]), so that the seminorm | · | o is submultiplicative.…”

“…In our exposition we shall essentially follow the nice presentation of the Banach algebra result given in [5], where the result is given directly in the non-commutative case (with an argument shorter than in [8]) and is based upon a simple lemma in analytic functions of one complex variable (reproved here as Lemma 2) produced for this purpose.…”

Invertibility preserving linear maps on ℒ(𝒳)

“…In the presence of commutativity, results of Gleason [8] and Kahane-Zelazko [11], refined by Zelazko [23], show that every unital invertibility preserving linear map from a Banach algebra A to a semisimple commutative Banach algebra B is multiplicative. (See also [16].…”