On the property $\mathit{IR}$ of Friis and Rørdam
| Autor: | Lawrence G. Brown |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Property (philosophy) 46L05 Rank (linear algebra) $C^{*}$-algebras 010102 general mathematics Cancellation property extension 0211 other engineering and technologies 021107 urban & regional planning 02 engineering and technology Extension (predicate logic) nonstable K-theory 01 natural sciences invertible law.invention Matrix (mathematics) Invertible matrix law 0101 mathematics Analysis Mathematics |
| Zdroj: | Banach J. Math. Anal. 13, no. 3 (2019), 599-611 |
| Popis: | Lin solved a longstanding problem as follows. For each $\epsilon \gt 0$ , there is $\delta \gt 0$ such that, if $h$ and $k$ are self-adjoint contractive $n\times n$ matrices and $\|hk-kh\|\lt \delta $ , then there are commuting self-adjoint matrices $h'$ and $k'$ such that $\|h'-h\|$ , $\|k'-k\|\lt \epsilon $ . Here $\delta $ depends only on $\epsilon $ and not on $n$ . Friis and Rørdam greatly simplified Lin’s proof by using a property they called $\mathit{IR}$ . They also generalized Lin’s result by showing that the matrix algebras can be replaced by any $C^{*}$ -algebras satisfying $\mathit{IR}$ . The purpose of this paper is to study the property $\mathit{IR}$ . One of our results shows how $\mathit{IR}$ behaves for $C^{*}$ -algebra extensions. Other results concern nonstable $K$ -theory. One shows that $\mathit{IR}$ (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. |
| Databáze: | OpenAIRE |
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