Trace spaces of simple nuclear C*-algebras with finite-dimensional extreme boundary

By Yasuhiko Sato. 2012

67 citations recorded by the source corpus.

Let A be a unital separable simple infinite-dimensional nuclear C*-algebra with at least one tracial state. We prove that if the trace space of A has compact finite-dimensional extreme boundary then there exist unital embeddings of matrix algebras into a certain central sequence algebra of A which is determined by the uniform topology on the trace space. As an application, it is shown that if furthermore A has strict comparison then A absorbs the Jiang-Su algebra tensorially.

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

View Paper: https://www.semanticscholar.org/paper/ba71098b1459523b4b3bdebb29d399fde03be5c3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. G'abor Szab'o
2. W. Winter
3. Stuart White
4. Aaron Tikuisis
5. Hannes Thiel
6. Yasuhiko Sato
7. J. Zacharias
8. Jianchao Wu

In Geometry and Topology

1. G'abor Szab'o
2. Stuart White
3. J. Zacharias
4. Hannes Thiel
5. Aaron Tikuisis
6. Yasuhiko Sato
7. W. Winter
8. N. Brown

In Mathematical Physics

1. G'abor Szab'o
2. W. Winter
3. Stuart White
4. Aaron Tikuisis
5. Hannes Thiel
6. J. Zacharias
7. Yasuhiko Sato
8. Jianchao Wu

Citations in the corpus

• Ranks of Operators in Simple $$C^*$$-algebras with Stable Rank One (2017, result): “The implication ‘(3)⇒(2)’ was shown to hold under the additional assumption that ∂eT (A) is compact and finite dimensional by independent works of KirchbergRørdam, [KR14], Sato, [Sat12], and Toms-White-Winter, [TWW15], extending the work of Matuia-Sato, [MS12], that covered the case of finitely many”
• STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN (2017, unknown): “In the six author paper Bosa, Brown, Sato, Tikuisis, White, and Winter [2015], Joan Bosa, Nate Brown, Yasuhiko Sato, Aaron Tikuisis, Stuart White and myself showed (ii) H) (i) when the tracial state space of A has compact extreme boundary; (ii) H) (i’) was shown assuming in addition that every trace”
• Rokhlin dimension for actions of residually finite groups (2014, background): “In view of the astonishing progess of C∗-algebra classification over the last few years [18, 13, 12, 73], which is largely driven by the discovery of various regularity properties (see for example [83, 81, 82, 59, 48, 36, 75, 65, 50, 84, 66, 5]), a natural and important question appears, which can b”
• Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, background): “, its extreme boundary ∂eT (A) is compact) and ∂eT (A) has finite covering dimension [31, 53, 60] (this extends the innovative work [38] which handles the case of finitely many extremal traces).”
• Nuclear dimension and $$\mathcal Z$$Z-stability (2014, background): “Moreover, the implications (i)⇒(ii) and (ii)⇒(iii) have been established in general by the third named author [47,49] (see [38] for an extension to the stably projectionless case) and Rørdam [31] respectively; recent progress of Matui and the first named author [25] establishes (iii)⇒(ii) for C∗-alg”