Infima in Cuntz semigroups and the structure of C*-algebras with stable rank one

Hannes Thiel (University of Muenster)

Thursday 10th May, 2018 16:00-17:00 Maths 311B


Let $A$ be a C*-algebra with stable rank one. We show that the Cuntz semigroup of $A$ satisfies Riesz interpolation. If $A$ is also separable, it follows that the Cuntz semigroup of $A$ has finite infima. This has several applications:

1. We confirm a conjecture of Blackadar and Handelman for unital C*-algebras with stable rank one: The (not necessarily lower semicontinuous) normalized dimension functions on $A$ form a Choquet simplex.

2. We confirm the global Glimm halving conjecture for unital C*-algebras with stable rank one: For each natural number $k$, the C*-algebra $A$ has no nonzero representations of dimension less than $k$ if and only if there exists a morphism from the cone over the algebra of $k$-by-$k$ matrices to $A$ with full range.

3. We solve the rank problem for separable, unital (not necessarily simple) C*-algebras with stable rank one that have no finite-dimensional 
quotients: For every lower semicontinuous, strictly positive, affine function $f$ on the Choquet simplex of normalized $2$-quasitraces on $A$, there exists a positive element in the stabilization of $A$ that has rank $f$. 

This is joint work with Ramon Antoine, Francesc Perera and Leonel Robert.

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