Gleason parts and point derivations for uniform algebras with dense invertible group

Izzo, Alexander J.

doi:10.1090/tran/7153

Cited by 7 publications

(24 citation statements)

References 20 publications

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“…The next result is a modification yielding a stronger conclusion from a stronger hypothesis. This generalizes [16,Theorem 4.1].…”

Section: Generalization Of the Construction Of Duval And Levenbergsupporting

confidence: 73%

Spaces with polynomial hulls that contain no analytic discs

Izzo

2019

Math. Ann.

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Extensions of the notions of polynomial and rational hull are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomial hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set in C 3 with a nontrivial polynomial hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and simple closed curves in C 4 with nontrivial polynomial hulls that contain no analytic discs. This answers a question raised by Bercovici in 2014 and can be regarded as a partial answer to a question raised by Wermer over 60 years ago. More generally, it is shown that every uncountable, compact subspace of a Euclidean space can be embedded as a subspace X of C N , for some N , in such a way as to have a nontrivial polynomial hull that contains no analytic discs. In the case when the topological dimension of the space is at most one, X can be chosen so as to have the stronger property that P (X) has a dense set of invertible elements.

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“…The next result is a modification yielding a stronger conclusion from a stronger hypothesis. This generalizes [16,Theorem 4.1].…”

Section: Generalization Of the Construction Of Duval And Levenbergsupporting

confidence: 73%

Spaces with polynomial hulls that contain no analytic discs

Izzo

2019

Math. Ann.

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“…Lemma 4.9 shows that P is not locally compact at 0 since P = L but 0 is not an interior point of P relative to L. Example 2. It is proven in [16,Theorem 1.5] that there exists a Swiss cheese L = D \ ( ∞ j=1 D j ) with a Gleason part P for R(L) of full measure in L. We claim that P is locally compact at no point. As shown in [16,Remark 5.6], the fact that P has full measure in the Swiss cheese L implies that P is dense in L. Applying Lemma 2.2 (for instance) shows that each point of ∞ j=1 ∂D j , a dense subset of L, is a one-point part, so P has empty interior in L. The claim now follows from Lemma 4.9.…”

Section: Gleason Parts In Polynomial Hullsmentioning

confidence: 87%

“…The proof of the next result is reminiscent of, though much simpler than, the proof of [16,Theorem 1.5].…”

Section: Gleason Parts In Polynomial Hullsmentioning

confidence: 99%

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Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

Izzo¹,

Papathanasiou²

2019

Can. J. Math.-J. Can. Math.

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We strengthen, in various directions, the theorem of Garnett that every σ-compact, completely regular space X occurs as a Gleason part for some uniform algebra. In particular, we show that the uniform algebra can always be chosen so that its maximal ideal space contains no analytic discs. We show that when the space X is metrizable, the uniform algebra can be chosen so that its maximal ideal space is metrizable as well. We also show that for every locally compact subspace X of a Euclidean space, there is a compact set K in some C N so that K \ K contains a Gleason part homeomorphic to X and K contains no analytic discs.

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“…Consequently, there is a compact polynomially convex set Y such that x 0 ∈ Y ⊂ Ω and m(Ω \ Y ) < ε/2. Using Theorem 1.3 and Lemma 2.3, we will obtain the desired arc by an argument similar to the proof of [8,Lemma 3.3]. Choose a countable collection {p j } of polynomials that is dense in P (Ω) and such that p j (x 0 ) = 0 for each j.…”

Section: The Proofsmentioning

confidence: 99%

Polynomial hulls of arcs and curves

Izzo

2020

Proc. Amer. Math. Soc.

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It is shown that there exist arcs and simple closed curves in C 3 with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded Runge domain of holomorphy in C N (N ≥ 2) there exist polynomially convex arcs and simple closed curves of almost full measure. These results, which strengthen earlier results of the author, are obtained as consequences of a general result about polynomial hulls of arcs and simple closed curves through Cantor sets.2010 Mathematics Subject Classification. Primary 32E20; Secondary 32A38, 32E30, 46J10, 46J15.

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Gleason parts and point derivations for uniform algebras with dense invertible group

Cited by 7 publications

References 20 publications

Spaces with polynomial hulls that contain no analytic discs

Spaces with polynomial hulls that contain no analytic discs

Topology of Gleason Parts in Maximal Ideal Spaces with no Analytic Discs

Polynomial hulls of arcs and curves

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