Quasidiagonality of nuclear C*-algebras
By Aaron Tikuisis, Stuart White, W. Winter. 2015
347 citations recorded by the source corpus.
We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C*-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms-Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C*-algebras.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/cd7d3ef439fade10c4a5844d0d52d44ea1fdaf53
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In Mathematical Physics
Citations in the corpus
- Nuclear dimension of simple stably projectionless C∗-algebras (2019, background): “In the unital case, a definitive answer for when K-theory and traces form a complete invariant is now known ([36, 48, 31, 22, 57, 69]).”
- Almost Finiteness and the Small Boundary Property (2018, background): “tiling argument in [6] (see the introduction to Section 8). Together with the Z-stability theorem from [21], work of Castillejos–Evington–Tikuisis–White–Winter [4], and the classification results from [12, 15, 38], this allows us to deduce our main result about C∗-crossed products: Theorem C (Theorem”
- Tensor products and regularity properties of Cuntz semigroups (2014, methodology): “This has been verified very recently in [TWW15].”
- Subalgebras of simple AF-algebras (2018, background): “Building on the work of Ozawa, Rørdam, and Sato in [62], it was shown in [82] that if G is a countable, discrete, amenable group, then Cr(G) embeds into an AF-algebra, although these results give no control over the codomain AF-algebra.”
- Sequentially split ∗-homomorphisms between C*-algebras (2015, background): “ty-type properties for C∗-algebras such as the universal coefficient theorem [53, 54], where it is not known whether it is automatic for nuclear C∗-algebras. In a very recent breakthrough by many hands [24, 14, 13, 62], it has been shown that these (a priori) different regularity-type properties are li”
- Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions (2016, background): “Given the increasingly clear parallels between concepts for vonNeumann algebras and for C∗-algebras, which has driven the C∗-algebraic theory to its recent breakthroughs [5,14,21,78], one would expect that there is much more to discover about amenable group actions on C∗-algebras.”
- Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, neutral): “Since the first version of this paper AT, SW and WW have shown in [82] that faithful traces on separable nuclear C∗-algebras in the UCT class are quasidiagonal.”
- Rokhlin dimension for actions of residually finite groups (2014, background): “Through recent groundbreaking progress in the Elliott programme by many hands, the combined main results of [13, 14, 19, 74] have completed the classification of separable, unital, simple C-algebras that satisfy the UCT and have finite nuclear dimension.”
- On the classification of simple amenable C*-algebras with finite decomposition rank (2015, background): “le (non-elementary) C*-algebra with finite nuclear dimension, which satisfies the UCT, and whose tracial states are all quasidiagonal, is in the class N1.This result, combined with the recent result of [29] (that the quasidiagonality hypothesis is redundant), yields that the class N1 includes all finit”