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Pervasive algebras and maximal subalgebras

Pamela Gorkin, Anthony G. O'Farrell (2011)

Studia Mathematica

A uniform algebra A on its Shilov boundary X is maximal if A is not C(X) and no uniform algebra is strictly contained between A and C(X). It is essentially pervasive if A is dense in C(F) whenever F is a proper closed subset of the essential set of A. If A is maximal, then it is essentially pervasive and proper. We explore the gap between these two concepts. We show: (1) If A is pervasive and proper, and has a nonconstant unimodular element, then A contains an infinite descending chain of pervasive...

Pervasive algebras on planar compacts

Jan Čerych (1999)

Commentationes Mathematicae Universitatis Carolinae

We characterize compact sets X in the Riemann sphere 𝕊 not separating 𝕊 for which the algebra A ( X ) of all functions continuous on 𝕊 and holomorphic on 𝕊 X , restricted to the set X , is pervasive on X .

Power boundedness in Banach algebras associated with locally compact groups

E. Kaniuth, A. T. Lau, A. Ülger (2014)

Studia Mathematica

Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. Pursuing our investigations of power bounded elements in B(G), we study the extension property for power bounded elements and discuss the structure of closed sets in the coset ring of G which appear as 1-sets of power bounded elements. We also show that L¹-algebras of noncompact motion groups and of noncompact IN-groups with polynomial growth do not share the so-called power boundedness property. Finally, we give a characterization...

Products of n open subsets in the space of continuous functions on [0,1]

Ehrhard Behrends (2011)

Studia Mathematica

Let O₁,...,Oₙ be open sets in C[0,1], the space of real-valued continuous functions on [0,1]. The product O₁ ⋯ Oₙ will in general not be open, and in order to understand when this can happen we study the following problem: given f₁,..., fₙ ∈ C[0,1], when is it true that f₁ ⋯ fₙ lies in the interior of B ε ( f ) B ε ( f ) for all ε > 0 ? ( B ε denotes the closed ball with radius ε and centre f.) The main result of this paper is a characterization in terms of the walk t ↦ γ(t): = (f₁(t),..., fₙ(t)) in ℝⁿ. It has to...

Proper uniform algebras are flat

R. C. Smith (2009)

Czechoslovak Mathematical Journal

In this brief note, we see that if A is a proper uniform algebra on a compact Hausdorff space X , then A is flat.

Properties of function algebras in terms of their orthogonal measures

Jan Čerych (1994)

Commentationes Mathematicae Universitatis Carolinae

In the present note, we characterize the pervasive, analytic, integrity domain and the antisymmetric function algebras respectively, defined on a compact Hausdorff space X , in terms of their orthogonal measures on X .

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