Left and right projections are equivalent in Rickart \(C^*\)-algebras
type: Item
instance of: scholarly article
title: Left and right projections are equivalent in Rickart \(C^*\)-algebras
zbMATH Open document ID: 0694.46038
DOI: 10.1016/0021-8693(89)90209-3
author: Pere Ara
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: \textit{I. Kaplansky} [Ann. Math., II. Ser. 53, 235-249 (1951; Zbl 0042.124)] conjectured that in every Rickert \(C^*\)-algebra left projections are equivalent to right projections. For finite Rickart \(C^*\)-algebras this was proved by \textit{D. Handelman} [Finite Rickart \(C^*\)-algebras and their properties, Adv. Math. Suppl. Studies 4, 171-196 (1979; Zbl 0511.46054)]. In the present paper the author constructs for any Rickart \(C^*\)- algebra A a closed ideal I(A) such that A/I(A) is a finite Rickart \(C^*\)-algebra and I(A) is contained in every ideal K such that A/K is finite. With the help of this and Handelman's result, the author proves the conjecture of Kaplansky in the general case.
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 4136778
zbMATH Keywords: Rickert \(C^*\)-algebra
zbMATH Keywords: left projections
reviewed by: Khristo N. Boyadzhiev
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90209-3
OpenAlex ID: W2093282092
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description: scientific article; zbMATH DE number 4136778