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Tracial oscillation zero and stable rank one
Part of:
Selfadjoint operator algebras
Published online by Cambridge University Press: 24 January 2024
Abstract
Let A be a separable (not necessarily unital) simple 
$C^*$-algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map 
$\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple 
$C^*$-algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map 
$\Gamma $ is surjective.
MSC classification
Secondary:
46L05: General theory of $C^*$-algebras
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
The first-named author was partially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant. The second-named author was partially supported by an NSF grant (DMS-1954600). Both authors would like to acknowledge the support during their visits to the Research Center of Operator Algebras at East China Normal University which is partially supported by Shanghai Key Laboratory of PMMP, Science and Technology Commission of Shanghai Municipality (STCSM), grant #22DZ2229014.
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