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Mat. Sb., 1999, Volume 190, Number 2, Pages 3–30 (Mi msb381)  

This article is cited in 1 scientific paper (total in 1 paper)

On the homotopy equivalence of simple AI-algebras

O. Yu. Aristov

Obninsk State Technical University for Nuclear Power Engineering

Abstract: Let $A$ and $B$ be simple unital AI-algebras (an AI-algebra is an inductive limit of $C^*$-algebras of the form $\bigoplus_i^kC([0,1],M_{N_i})$. It is proved that two arbitrary unital homomorphisms from $A$ into $B$ such that the corresponding maps $\mathrm K_0A\to\mathrm K_0B$ coincide are homotopic. Necessary and sufficient conditions on the Elliott invariant for $A$ and $B$ to be homotopy equivalent are indicated. Moreover, two algebras in the above class having the same $\mathrm K$-theory but not homotopy equivalent are constructed. A theorem on the homotopy of approximately unitarily equivalent homomorphisms between AI-algebras is used in the proof, which is deduced in its turn from a generalization to the case of AI-algebras of a theorem of Manuilov stating that a unitary matrix almost commuting with a self-adjoint matrix $h$ can be joined to 1 by a continuous path consisting of unitary matrices almost commuting with $h$.

DOI: https://doi.org/10.4213/sm381

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English version:
Sbornik: Mathematics, 1999, 190:2, 165–191

Bibliographic databases:

UDC: 517.986.32
MSC: Primary 46L85, 58B30; Secondary 46L89
Received: 14.05.1998

Citation: O. Yu. Aristov, “On the homotopy equivalence of simple AI-algebras”, Mat. Sb., 190:2 (1999), 3–30; Sb. Math., 190:2 (1999), 165–191

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. Korchagin A.I., “Topological Complexity of Certain Classes of $C^*$-Algebras”, Russ. J. Math. Phys., 24:3 (2017), 347–353  crossref  mathscinet  zmath  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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