Fibers over the sphere of a uniformly convex Banach space
type: Item
instance of: scholarly article
title: Fibers over the sphere of a uniformly convex Banach space
zbMATH Open document ID: 0976.46037
author: Jeff D. Farmer
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: Let \(B\) be the open unit ball of an infinite-dimensional complex Banach space \(X\). In this paper the author studies the boundary behavior of bounded analytic functions on \(B\) when \(X\) is uniformly convex. Several interesting results are obtained regarding the structure of the spectrum of \(H^\infty (B)\). Among them: If \(X\) is uniformly convex, and \((\varphi_i)\) lie in fibers over the boundary and converge to \(\varphi\) in the Gelfand topology, then all but finitely many \(\varphi_i\) lie in the same fiber as \(\varphi\). If \(X=\ell_p\), there are uniformly homeomorphic copies of balls of ultrapowers of \(\ell_p\) in all fibers.
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 46G20
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 1548271
zbMATH Keywords: bounded analytic functions
zbMATH Keywords: spectrum
zbMATH Keywords: boundary behavior
reviewed by: Ignacio Zalduendo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132179
OpenAlex ID: W1979374838
DOI: 10.1307/MMJ/1030132179
description: scientific article; zbMATH DE number 1548271