Holomorphic functions with large cluster sets

Alves, Thiago R.; Carando, Daniel

doi:10.1002/mana.201900238

Cited by 2 publications

(2 citation statements)

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“…We come back to the sets just defined with respect to the algebra H ∞ (D X ). Having in mind that an image of a strongly c-algebrable set under an isometric isomorphism is again strongly c-algebrable and that τ * (ρB ∞ ) = ρτ * (B ∞ ) = ρD σ X for 0 < ρ < 1, we get the following result by combining Proposition 5.6, 5.7 with the above results in [1].…”

Section: By Proposition 53 We Have Thatmentioning

confidence: 80%

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The spectra of Banach algebras of holomorphic functions on polydisk type domains

Choi¹,

Jung²,

Maestre³

2020

Preprint

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R.M. Aron et al. [3] presented that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra H ∞ (B c0 ). On the other hand, B.J. Cole and T.W. Gamelin [15] showed that) in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets U of a Banach space X for which H ∞ (U ) is isometrically isomorphic to H ∞ (B c0 ). We prove that there exist polydisk type domains U of any infinite dimensional Banach space X with a Schauder basis such that H ∞ (U ) is isometrically isomorphic to H ∞ (B c0 ), which also generalizes the result by Cole and Gamelin [15]. We also show that the Cluster Value Theorem is true for H ∞ (U ). As the dual space X * is not necessarily contained in H ∞ (U ) for such a domain U , we find a natural way to define fibers of the spectrum of H ∞ (U ), and study the analytic and algebraic structure of its spectrum including its Gleason parts.

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Section: By Proposition 53 We Have Thatmentioning

confidence: 80%

“…Recently, T. R. Alves and D. Carando [1] studied algebraic structure in the set of holomorphic functions which have large cluster sets at every point. Given a Banach space X and a set M ⊂ B X * * , they defined the following sets:…”

Section: By Proposition 53 We Have Thatmentioning

confidence: 99%