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Closed ideals in a new class of algebras of holomorphic functions on the disc

Merino-Cruz H., Wawrzyńczyk A.

Commentationes Mathematicae

2014

Vol 54, No. 1

29--37

We define a new class of Banach algebras of holomorphic functions on the unit disc D which contains the algebras studied in [GMR2] and [GW]. To a function G of the class A1(D) nowhere vanishing in D we associate a Banach algebra AnG(D) contained in the disc algebra A(D). We prove that all closed ideals of AnG(D) of at most countable hull are of the standard form.

Ditkin's condition and ideals with at most countable hull in algebras of functions analytic in the unit disc

Sołtysiak A., Wawrzyńczyk A.

Commentationes Mathematicae

2012

Vol. 52, [Z] 1

101-112

Schur lemma and the spectral mapping formula

Wawrzyńczyk A.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 1

63-69

Let B be a complex topological unital algebra. The left joint spectrum of a set S ⊂ B is defined by the formula σl(S) = {(λ(s))s∈s ∈ C[sup]s | {s - λ(s)}s∈s generates a proper left ideal}. Using the Schur lemma and the Gelfand-Mazur theorem we prove that σl(S) has the spectral mapping property for sets S of pairwise commuting elements if (i) B is an m-convex algebra with all maximal left ideals closed, or (ii) B is a locally convex Waelbroeck algebra. The right ideal version of this result is also valid.