2 results
Tracial oscillation zero and stable rank one
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- Journal:
- Canadian Journal of Mathematics , First View
- Published online by Cambridge University Press:
- 24 January 2024, pp. 1-68
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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.
ON POSSIBLE VALUES OF THE INTERIOR ANGLE BETWEEN INTERMEDIATE SUBALGEBRAS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 117 / Issue 1 / August 2024
- Published online by Cambridge University Press:
- 17 July 2023, pp. 44-66
- Print publication:
- August 2024
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We show that all values in the interval
$[0,{\pi }/{2}]$ can be attained as interior angles between intermediate subalgebras (as introduced by Bakshi and the first named author [‘Lattice of intermediate subalgebras’, J. Lond. Math. Soc. (2) 104(2) (2021), 2082–2127]) of a certain inclusion of simple unital 
$C^*$-algebras. We also calculate the interior angles between intermediate crossed product subalgebras of any inclusion of crossed product algebras corresponding to any action of a countable discrete group and its subgroups on a unital 
$C^*$-algebra.
