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Three spectral notions for representations of commutative Banach algebras

Yngve Domar, Lars-Ake Lindahl (1975)

Annales de l'institut Fourier

Let T be a bounded representation of a commutative Banach algebra B . The following spectral sets are studied. Λ 1 ( T ) : the Gelfand space of the quotient algebra B / Ker T . Λ 2 ( T ) : the Gelfand space of the operator algebra Im T . Λ 3 ( T ) : those characters φ of B for which the inequalities ∥ T b x - b ^ ( φ ) x ∥ < ϵ ∥ x ∥ , b ∈ F , have a common solution x ≠ 0 , for any ϵ > 0 and any finite subset F of B . A theorem of Beurling on the spectrum of L ∞ -functions and results of Slodkowski and Zelazko on joint topological divisors of zero appear as special cases of our theory by...

Topological algebras with an orthogonal total sequence

Hermann Render (1997)

Colloquium Mathematicae

The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence ( x n ) n ∈ ℕ satisfying x n 2 = x n and x n x n + 1 = x n + 1 for all n ∈ ℕ.

Topological algebras with maximal regular ideals closed

Mati Abel (2012)

Open Mathematics

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.

Topologies on the space of ideals of a Banach algebra

Ferdinand Beckhoff (1995)

Studia Mathematica

Some topologies on the space Id(A) of two-sided and closed ideals of a Banach algebra are introduced and investigated. One of the topologies, namely τ ∞ , coincides with the so-called strong topology if A is a C*-algebra. We prove that for a separable Banach algebra τ ∞ coincides with a weaker topology when restricted to the space Min-Primal(A) of minimal closed primal ideals and that Min-Primal(A) is a Polish space if τ ∞ is Hausdorff; this generalizes results from [1] and [5]. All subspaces of Id(A)...

Trivial generators for nontrivial fibres

Linus Carlsson (2008)

Mathematica Bohemica

Pseudoconvex domains are exhausted in such a way that we keep a part of the boundary fixed in all the domains of the exhaustion. This is used to solve a problem concerning whether the generators for the ideal of either the holomorphic functions continuous up to the boundary or the bounded holomorphic functions, vanishing at a point in ℂ n where the fibre is nontrivial, has to exceed n . This is shown not to be the case.

Trivial Jensen measures without regularity

J. F. Feinstein (2001)

Studia Mathematica

In this note we construct Swiss cheeses X such that R(X) is non-regular but such that R(X) has no non-trivial Jensen measures. We also construct a non-regular uniform algebra with compact, metrizable character space such that every point of the character space is a peak point.

Currently displaying 761 – 780 of 914