\(C^*\)-algebras of stable rank one and their Cuntz semigroups. (English) Zbl 1490.19005

Antoine, Ramon; Perera, Francesc; Robert, Leonel; Thiel, Hannes

Duke Math. J. 171, No. 1, 33-99 (2022).

The Cuntz semigroup is a rather fine invariant of a \(C^*\)-algebra. It is possible that it may be used to classify simple \(C^*\)-algebras which are not yet classified by the Elliott invariant. Regularity properties of the Cuntz semigroup may be related to classifiability by the Elliott invariant. Whereas \(K\)-theory only looks at equivalence classes of projections in matrix algebras over a given \(C^*\)-algebra, the Cuntz semigroup looks at equivalence classes of all positive elements, with a suitable equivalence relation that is a version of Murray-von Neumann equivalence up to \(\epsilon\).
This article studies regularity properties of the Cuntz semigroup in case the underlying \(C^*\)-algebra has stable rank one. For such \(C^*\)-algebras, the Cuntz semigroup is shown to have the Riesz interpolation property. Assuming separability as well, every pair of elements of the Cuntz semigroup has an infimum, and addition is distributive over the infimum operation. For a unital \(C^*\)-algebra of stable rank one, it is shown that its dimension functions form a Choquet simplex, as conjectured by Blackadar and Handelman for all \(C^*\)-algebras. In addition, \(K_0^*(A)\), the Grothendieck group of the Cuntz semigroup, is shown to be an interpolation group.
In case no subquotient of the \(C^*\)–algebra is an elementary\(C^*\)–algebra, it is characterised which functions on the space of quasitraces may be written as ranks of Cuntz semigroup elements. The global Glimm halving problem also has a positive answer for \(C^*\)-algebras of stable rank one. It produces a *-homomorphism with full range to the \(C^*\)-algebra whose domain is a cone over a matrix algebra \(M_k\), provided the \(C^*\)-algebra has no finite-dimensional rpresentations of dimension less than \(k\). In particular, if the \(C^*\)-algebra has no finite-dimensional representations at all, then it receives *-homomorphisms with full range from \(M_k(C_0((0,1]))\) for all \(k\). Finally, the article characterises when the Cuntz semigroup of a simple, unital, separable \(C^*\)-algebra of stable rank one has finite radius of comparison or \(m\)-comparision for some natural number \(m\) or local comparison.
Some of the results are proven for any ordered semigroup \(S\) with certain properties that Cuntz semigroups of \(C^*\)-algebras always have. The space of quasitraces on the Cuntz semigroup has an analogue \(F(S)\) in this generality, and elements of \(S\) induce functions on \(F(S)\) with certain properties. The article finds a sufficient criterion for the map from \(S\) to functions on \(F(S)\) to preserve infima.

Reviewer: Ralf Meyer (Göttingen)

MSC:

19K14	\(K_0\) as an ordered group, traces
46L35	Classifications of \(C^*\)-algebras
46L08	\(C^*\)-modules

Keywords:

\(C^*\)-algebra; Cuntz semigroup; Hilbert \(C^*\)-module; semilattice; stable rank one; weak comparison; inf-semilattice

References:

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