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Nowhere scattered multiplier algebras

Published online by Cambridge University Press:  05 January 2024

Eduard Vilalta Open the ORCID record for Eduard Vilalta [Opens in a new window]
Eduard Vilalta*
Affiliation:
Fields Institute for Research in Mathematical Sciences, Toronto M5T 3J1, Canada Previous institution: Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Barcelona 08193, Spain (eduardvilaltavila@gmail.com)
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Abstract

We study sufficient conditions under which a nowhere scattered $\mathrm {C}^*$-algebra $A$ has a nowhere scattered multiplier algebra $\mathcal {M}(A)$, that is, we study when $\mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $\sigma$-unital $\mathrm {C}^*$-algebra $A$ of

  1. (i) finite nuclear dimension, or

  2. (ii) real rank zero, or

  3. (iii) stable rank one with $k$-comparison,

is nowhere scattered if and only if $\mathcal {M}(A)$ is.

Keywords

C*-algebramultiplier algebraCuntz semigroup

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 4 , August 2025 , pp. 1113 - 1142
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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