Ranks of Operators in Simple $$C^*$$-algebras with Stable Rank One

By Hannes Thiel. Communications in Mathematical Physics, 2017

71 citations recorded by the source corpus.

Let $A$ be a separable, unital, simple C*-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of $A$ is realized as the rank of an operator in the stabilization of $A$. Assuming moreover that $A$ has locally finite nuclear dimension, we deduce that $A$ is $\mathcal{Z}$-stable if and only if it has strict comparison of positive elements. In particular, the Toms-Winter conjecture holds for separable, unital, simple, approximately subhomogeneous C*-algebras with stable rank one.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a54975c36b629f964a4bd60b0c19ae5651b7c012

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In Algebra and Number Theory

1. W. Winter
2. G'abor Szab'o
3. Andrew S. Toms
4. Stuart White
5. Z. Niu
6. Yasuhiko Sato
7. J. Zacharias
8. Aaron Tikuisis

In Geometry and Topology

1. W. Winter
2. G'abor Szab'o
3. Z. Niu
4. David Kerr
5. Stuart White
6. Yasuhiko Sato
7. Huaxin Lin
8. Andrew S. Toms

In Mathematical Physics

1. W. Winter
2. G'abor Szab'o
3. Andrew S. Toms
4. Stuart White
5. David Kerr
6. Yasuhiko Sato
7. Z. Niu
8. J. Zacharias

Citations in the corpus

• STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN (2017, background): “Hannes Thiel (2017).”
• C*-algebras of stable rank one and their Cuntz semigroups (2018, unknown): “. Further, this semilattice structure is compatible with addition; see Theorem 3.8. In the sequel, we write x 1,x 2 6 y 1,y 2 to mean xi 6 yj for i,j = 1,2. 3.1. The following axiom was introduced in [Thi17]. A Cu-semigroup S is said to satisfy axiom (O6+) if for every a,b,c,x′,x,y′,y ∈ S satisfying”