Decomposition rank of UHF-absorbing c* -algebras

By H. Matui, Yasuhiko Sato. 2013

152 citations recorded by the source corpus.

Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/562b1bcc3b3ae22ca1e5340220eedbaf7b0ea660

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Citations in the corpus

Almost Finiteness and the Small Boundary Property (2018, background): “ith the topological is very much 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 [31, 32]. One of the main aims of the present paper is to push this technical connection between mea”
UCT-Kirchberg algebras have nuclear dimension one (2014, unknown): “Matui and Sato [13] subsequently improved the estimate for simple Kirchberg algebras from 5 to 3, and their result is valid for non-UCT Kirchberg algebras, if any exist; and Barlak, Enders, Matui, Szab´o and Winter have showed how to recover the Matui-Sato estimate from a general relationship…”
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”
Cartan subalgebras in C*-algebras. Existence and uniqueness (2017, background): “Classification of C*algebras has seen tremendous advances recently [66, 67, 34, 35, 55, 22, 14, 62].”
Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions (2016, methodology): “Let us recall a useful well-known fact, which was implicitely used for example in [2] and [56].”
STRUCTURE OF NUCLEAR C*-ALGEBRAS: FROM QUASIDIAGONALITY TO CLASSIFICATION AND BACK AGAIN (2017, unknown): “Dusa McDuff (1970).”
Nuclear dimension and $$\mathcal Z$$Z-stability (2014, methodology): “We aim to use such maps in place of the ∗-homomorphisms used in [9,26].”
Subalgebras of simple AF-algebras (2018, background): “The minimal route through the literature seems to be the quasidiagonality theorem of [82], the results of Matui-Sato to show A is tracially AF [55], and the classification of separable, simple, unital, nuclear, tracially AF-algebras satisfying the UCT due to Lin in [50].”
Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, methodology): “Indeed, all Kirchberg algebras have finite nuclear dimension ([97, 57, 74]), while the exotic examples of simple, infinite but not purely infinite C∗-algebras from [72] cannot have finite nuclear dimension.”
Quasidiagonality of nuclear C*-algebras (2015, background): “This is the first abstract result of this kind (although quasidiagonality was identified as the differentiating feature between nuclear dimension and decomposition rank in [53], particularly in light of [66]).”
On the classification of simple amenable C*-algebras with finite decomposition rank (2015, background): “In [20], Matui and Sato showed that A⊗ UHF can be tracially approximated by finite dimensional C*-algebras (i.”