Ideals and structure of operator algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1261.47090
author: Melahat Almus
author: David P. Blecher
author: Sonia Sharma
publication date: 2012-06-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.3103
Mathematics Subject Classification ID: 47L25
Mathematics Subject Classification ID: 47L30
Mathematics Subject Classification ID: 47L40
Mathematics Subject Classification ID: 46H10
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 6045975
zbMATH Keywords: nonselfadjoint operator algebra
zbMATH Keywords: one-sided ideals
zbMATH Keywords: hereditary subalgebra
zbMATH Keywords: matrix units
zbMATH Keywords: minimal ideals
zbMATH Keywords: annihilator algebra
zbMATH Keywords: Haagerup tensor product
MaRDI profile type: Publication
arXiv ID: 0907.3103
arXiv classification: math.OA
arXiv classification: math.FA
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title: Ideals and structure of operator algebras
published in: Journal of Operator Theory
review text: An operator algebra is a norm closed algebra of operators on a Hilbert space. The paper under review continues the program of the authors of studying the structure of operator algebras and operator spaces using one-sided ideals of special types. In particular, \(r\)-ideals are right ideals of an operator algebra possessing a left contractive approximate identity. Swapping ``left'' and ``right'' we get the definition of \(l\)-ideal.NEWLINENEWLINE After recording many general results about one-sided ideals in Section 2, authors dedicate Section 3 to the existence of non-trivial \(r\)-ideals. Section 4 introduces the concept of a ``matricial'' algebra, that is, having a full set of matrix units whose span is dense in the algebra. In particular, there is a Wedderburn type structure theorems for operator algebras, using \(1\)-matricial algebras as the building blocks. Section 5 continues the development and presents some new characterizations of \(C^*\)-algebras consisting of compact operators. The final section considers the ideal structure of the Haagerup tensor product of operator algebras.
reviewed by: Vladimir V. Kisil
description: scientific article; zbMATH DE number 6045975