Uniqueness of the maximal ideal of the Banach algebra of bounded operators on \(C([0, \omega_1])\)
type: Item
instance of: scholarly article
title: Uniqueness of the maximal ideal of the Banach algebra of bounded operators on \(C([0, \omega_1])\)
zbMATH Open document ID: 1253.46026
author: Tomasz Kania
author: Niels Jakob Laustsen
published in: Journal of Functional Analysis
publication date: 2012-06-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.4800
Mathematics Subject Classification ID: 46B28
Mathematics Subject Classification ID: 46H10
zbMATH DE Number: 6044517
zbMATH Keywords: continuous functions on the first uncountable ordinal
zbMATH Keywords: bounded operators on Banach spaces
zbMATH Keywords: maximal ideal
Wikidata QID: Q107657544
reviewed by: Elói M. Galego
MaRDI profile type: Publication
OpenAlex ID: W2024870381
arXiv ID: 1112.4800
cites work: The structure of weakly compact sets in Banach spaces
cites work: The Banach space S is complementably minimal and subsequentially prime
cites work: COMPLEMENTATION AND EMBEDDINGS OF c0(I) IN BANACH SPACES
cites work: Q3879732
cites work: Asplund operators and the Szlenk index
cites work: Commutators on (Σlq)p
cites work: Commutators on ℓ∞
cites work: Commutators on 𝐿_{𝑝}, 1≤𝑝<∞
cites work: On the Primariness of the Banach Space l ∞ /C 0
cites work: Banach space theory. The basis for linear and nonlinear analysis
cites work: Various slicing indices on Banach spaces
cites work: Biorthogonal systems in Banach spaces
cites work: Norm closed operator ideals in Lorentz sequence spaces
cites work: Banach spaces whose algebra of bounded operators has the integers as their \(K_0\)-group
cites work: Q4679516
cites work: The lattice of closed ideals in the Banach algebra of operators on certain Banach spaces
cites work: Dichotomy theorems for random matrices and closed ideals of operators on \((\bigoplus _{n=1}^{\infty }\ell _{1}^{n})_{c_o}\)
cites work: Q3417118
cites work: Norm closed ideals in the algebra of bounded linear operators on Orlicz sequence spaces
cites work: Q5640213
cites work: Q4136031
cites work: Continuity of Derivations on B(E) for Certain Banach Spaces E
cites work: Homomorphisms from ℬ (ℬ(ωη ))
cites work: On C(S)-subspaces of separable Banach spaces
cites work: On factors of C(\([0,1]\)) with non-separable dual
cites work: Continuous Functions on Compact Spaces Without Perfect Subsets
cites work: Q3676657
cites work: Q3266617
cites work: The non-existence of a separable reflexive Banach space universal for all separable reflexive Banach spaces
cites work: Strictly Singular Operators and Their Conjugates
DOI: 10.1016/J.JFA.2012.03.011
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review text: Denote by \(C([0, \omega_{1}])\) the Banach space of continuous functions on the ordinal interval \([0, \omega_{1}]\), where \(\omega_{1}\) is the first uncountable ordinal. Denote also by \({\mathcal B}(C([0, \omega_{1}]))\) the Banach algebra of bounded operators on \(C([0, \omega_{1}])\). The starting point of the paper under review is a result of \textit{W. Rudin} [Proc. Am. Math. Soc. 8, 39--42 (1957; Zbl 0077.31103)] which implies that each \(T\) in \({\mathcal B}(C([0, \omega_{1}]))\) has a natural representation as a \([0, \omega_{1}] \times [0, \omega_{1}]\)-matrix. Then, it is recalled that \textit{R. J. Loy} and \textit{G. A. Willis} [J. Lond. Math. Soc., II. Ser. 40, No. 2, 327--346 (1989; Zbl 0651.47035)] observed that the set of operators whose final column is continuous when viewed as scalar-valued functions on \([0, \omega_{1}]\) defines a maximal ideal of codimension one in \({\mathcal B}(C([0, \omega_{1}]))\).NEWLINENEWLINEThus, the authors of the present paper give a coordinate-free characterization of this ideal and deduce from it that \({\mathcal B}(C([0, \omega_{1}]))\) contains no other maximal ideals. Then, they obtain a list of equivalent conditions describing the strictly smaller ideal of operators with separable range. They also investigate the structure of the lattice of all closed ideals of \({\mathcal B}(C([0, \omega_{1}]))\).
description: scientific article; zbMATH DE number 6044517