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Classification of homomorphisms from $C(\Omega)$ to a $C^*$-algebra

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  21 April 2025

Qingnan An ,
George Arthur Elliott  and
Zhichao Liu Open the ORCID record for Zhichao Liu [Opens in a new window]
Qingnan An
Affiliation:
School of Mathematics and Statistics, Northeast Normal University, Changchun, China e-mail: qingnanan1024@outlook.com
George Arthur Elliott
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ON, Canada e-mail: elliott@math.toronto.edu
Zhichao Liu*
Affiliation:
School of Mathematical Sciences, Dalian University of Technology, Dalian, China
*
e-mail: lzc.12@outlook.com
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Abstract

Let $\Omega $ be a compact subset of $\mathbb {C}$ and let A be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero, and strict comparison. We show that, given a Cu-morphism ${\alpha :\mathrm { Cu}(C(\Omega ))\to \mathrm {Cu}(A)}$ with , there exists a homomorphism $\phi : C(\Omega )\to A$ such that $\mathrm {Cu}(\phi )=\alpha $. Moreover, if $K_1(A)$ is trivial, then $\phi $ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^*$-algebras to A in terms of the Cuntz semigroup.

Keywords

ClassificationCuntz semigroupstrict comparison

MSC classification

Primary: 46L35: Classifications of $C^*$-algebras 46L05: General theory of $C^*$-algebras

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 24
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The research of the first author was supported by NNSF of China (Grant No.: 12101113). The research of the second author was supported by NSERC of Canada. The third author was supported by LiaoNing Revitalization Talents Program (No.: XLYC2403058) and NNSF of China (Grant No.: 12101102).

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