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July 2019 On the property IR of Friis and Rørdam
Lawrence G. Brown
Banach J. Math. Anal. 13(3): 599-611 (July 2019). DOI: 10.1215/17358787-2019-0004

Abstract

Lin solved a longstanding problem as follows. For each ϵ>0, there is δ>0 such that, if h and k are self-adjoint contractive n×n matrices and hkkh<δ, then there are commuting self-adjoint matrices h' and k' such that h'h, k'k<ϵ. Here δ depends only on ϵ and not on n. Friis and Rørdam greatly simplified Lin’s proof by using a property they called IR. They also generalized Lin’s result by showing that the matrix algebras can be replaced by any C-algebras satisfying IR. The purpose of this paper is to study the property IR. One of our results shows how IR behaves for C-algebra extensions. Other results concern nonstable K-theory. One shows that IR (at least the stable version) implies a cancellation property for projections which is intermediate between the strong cancellation satisfied by C-algebras of stable rank 1 and the weak cancellation defined in a 2014 paper by Pedersen and the author.

Citation

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Lawrence G. Brown. "On the property IR of Friis and Rørdam." Banach J. Math. Anal. 13 (3) 599 - 611, July 2019. https://doi.org/10.1215/17358787-2019-0004

Information

Received: 25 September 2018; Accepted: 3 January 2019; Published: July 2019
First available in Project Euclid: 31 May 2019

zbMATH: 07083763
MathSciNet: MR3978939
Digital Object Identifier: 10.1215/17358787-2019-0004

Subjects:
Primary: 46L05

Keywords: $C^{*}$-algebras , Extension‎ , invertible , nonstable K-theory

Rights: Copyright © 2019 Tusi Mathematical Research Group

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Vol.13 • No. 3 • July 2019
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