The density of peak points in the Shilov boundary of a Banach function algebra

Honary, Taher Ghasemi

doi:10.1090/s0002-9939-1988-0943070-9

Cited by 3 publications

(3 citation statements)

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“…E.Bishop proved in [Bis59] that, if the compact Hausdorff space is assumed to be metrizable, then the closure of the set of peak points and the Shilov boundary for uniform subalgebras of continuous functions coincide. This is also true for any Banach subalgebra of continuous functions due to the results of H.G.Dales [Dal71] (see also [Hon88]). Note that using upper semi-continuous functions similar identities were obtained in [Sic62] and [Wit83].…”

Section: Introductionmentioning

confidence: 72%

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Bergman–Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Pawlaschyk

2016

Ann. Polon. Math.

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We introduce a notion of the Shilov boundary for some subclasses of upper semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of that subclass attain their maximum. For certain subclasses with simple structure one can show the existence and uniqueness of the Shilov boundary. Then we provide its relation to the set of peak points and establish Bishop-type theorems. As an application we obtain a generalization of Bychkov's theorem which gives a geometric characterization of the Shilov boundary for q-plurisubharmonic functions on convex bounded domains. In the case of bounded pseudoconvex domains with smooth boundary we also show that some parts of the Shilov boundary for q-plurisubharmonic functions are foliated by q-dimensional complex submanifolds.

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Section: Introductionmentioning

confidence: 72%

“…We recall Bishop's theorem. Further generalizations can be found in [Sic62], [Dal71] and [Hon88]. Proof.…”

Section: Proposition 34mentioning

confidence: 99%

Bergman–Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Pawlaschyk

2016

Ann. Polon. Math.

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“…Theorem 36 (see [11,Theorem 2.3] and [21]). Let be a Banach function algebra on a compact metrizable space .…”

Section: Theorem 34 Let Be a Real Uniform Function Algebra On ( ) Then Every Peak Set For Contains A --Point Formentioning

confidence: 99%

Choquet and Shilov Boundaries, Peak Sets, and Peak Points for Real Banach Function Algebras

Alimohammadi

Honary

2013

Journal of Function Spaces and Applications

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Let be a compact Hausdorff space and let be a topological involution on . In 1988, Kulkarni and Arundhathi studied Choquet and Shilov boundaries for real uniform function algebras on ( , ). Then in 2000, Kulkarni and Limaye studied the concept of boundaries and Choquet sets for uniformly closed real subspaces and subalgebras of ( , ) or ( ). In 1971, Dales obtained some properties of peak sets and p-sets for complex Banach function algebras on . Later in 1990, Arundhathi presented some results on peak sets for real uniform function algebras on ( , ). In this paper, while we present a brief account of the work of others, we extend some of their results, either to real subspaces of ( , ) or to real Banach function algebras on ( , ).

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Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

Hatori

Miura

Takagi

2007

Journal of Mathematical Analysis and Applications

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Let T be a surjective map from a unital semi-simple commutative Banach algebra A onto a unital commutative Banach algebra B. Suppose that T preserves the unit element and the spectrum σ (fg) of the product of any two elements f and g in A coincides with the spectrum σ (Tf T g). Then B is semi-simple and T is an isomorphism. The condition that T is surjective is essential: An example of a non-linear and nonmultiplicative unital map from a commutative C*-algebra into itself such that σ (Tf T g) = σ (fg) holds for every f, g are given. We also show an example of a surjective unital map from a commutative C*-algebra onto itself which is neither linear nor multiplicative such that σ (Tf T g) ⊂ σ (fg) holds for every f, g.

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The density of peak points in the Shilov boundary of a Banach function algebra

Cited by 3 publications

References 3 publications

Bergman–Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Bergman–Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Choquet and Shilov Boundaries, Peak Sets, and Peak Points for Real Banach Function Algebras

Unital and multiplicatively spectrum-preserving surjections between semi-simple commutative Banach algebras are linear and multiplicative

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