Strict comparison and $$ \mathcal{Z} $$-absorption of nuclear C∗-algebras

By H. Matui, Yasuhiko Sato. 2011

83 citations recorded by the source corpus.

For any unital separable simple infinite-dimensional nuclear C∗-algebra with finitely many extremal traces, we prove that $$ \mathcal{Z} $$-absorption, strict comparison and property (SI) are equivalent. We also show that any unital separable simple nuclear C∗-algebra with tracial rank zero is approximately divisible, and hence is $$ \mathcal{Z} $$-absorbing.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f229e668e2a88ff4d6851daa59cf772d080b5cfb

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In Applied Mathematics

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

In Mathematical Physics

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

Citations in the corpus

• Almost Finiteness and the Small Boundary Property (2018, methodology): “This commingling of the measure-theoretic with the topological is very much in line with the kind of C∗-application of von Neumann algebra technology that was pioneered by Matui and Sato in their groundbreaking work on the Toms–Winter conjecture [35, 36].”
• Dixmier approximation and symmetric amenability for C*-algebras (2013, result): “g– Rørdam ([KR]), Sato ([Sa2]), and Toms–White–Winter ([TWW]), who studied the central sequence algebra of a C∗-algebra modulo uniformly 2-norm null sequences, in order to extend Matui–Sato’s result ([MS]) from C∗-algebras with finitely many extremal tracial states to more general ones. In fact, our ”
• Nuclear dimension and $$\mathcal Z$$Z-stability (2014, background): “stablished 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∗-algebras with a unique tracial state (this has subsequently been ”
• Rokhlin dimension for actions of residually finite groups (2014, background): “me. 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 ca”
• STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN (2017, background): “ is Z-stable if and only if it is O∞-stable; see [45]. In the finite case, (ii) =⇒ (iii) was shown by Rørdam in [45]. I showed first (i’) =⇒ (ii) and then (i) =⇒ (ii) in [60] and [61], respectively. In [36], Matui and Sato showed (iii) =⇒ (ii) when Ahas only one tracial state. In each of [48], [31], a”
• Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, background): “ult of [89]. DIMENSION AND 2-COLOURED CLASSIFICATION 3 (i.e., it is nonempty and its extreme boundary ∂eT(A) is compact) and ∂eT(A) has finite covering dimension ([46, 71, 80]) — these extend the work [54] which handles the case of finitely many extremal traces. In more recent breakthroughs, (ii) =⇒ (”