Nuclear dimension and -stability of pure C ∗ -algebras

By W. Winter. 2010

217 citations recorded by the source corpus.

In this article I study a number of topological and algebraic dimension type properties of simple C*-algebras and their interplay. In particular, a simple C*-algebra is defined to be (tracially) (m,\bar{m})-pure, if it has (strong tracial) m-comparison and is (tracially) \bar{m}-almost divisible. These notions are related to each other, and to nuclear dimension. The main result says that if a separable, simple, nonelementary, unital C*-algebra A with locally finite nuclear dimension is (m,\bar{m})-pure, then it absorbs the Jiang-Su algebra Z tensorially. It follows that A is Z-stable if and only if it has the Cuntz semigroup of a Z-stable C*-algebra. The result may be regarded as a version of Kirchberg's celebrated theorem that separable, simple, nuclear, purely infinite C*-algebras absorb the Cuntz algebra O_\infty tensorially. As a corollary we obtain that finite nuclear dimension implies Z-stability for separable, simple, nonelementary, unital C*-algebras; this settles an important case of a conjecture by Toms and the author. The main result also has a number of consequences for Elliott's program to classify nuclear C*-algebras by their K-theory data. In particular, it completes the classification of simple, unital, approximately homogeneous algebras with slow dimension growth by their Elliott invariants, a question left open in the Elliott-Gong-Li classification of simple AH algebras. Another consequence is that for simple, unital, approximately subhomogeneous algebras, slow dimension growth and Z-stability are equivalent. In the case where projections separate traces, this completes the classification of simple, unital, approximately subhomogeneous algebras with slow dimension growth by their ordered K-groups.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/c15f1d71b8ec4f4b3ccaf2b425a7521ee5e2ea80

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Citations in the corpus

Covering Dimension of C*-Algebras and 2-Coloured Classification (2015, methodology): “The implications (i) =⇒ (ii) and (ii) =⇒ (iii) of Conjecture A were proved by WW and by Rørdam, respectively ([92, 94, 73]), while (iii) =⇒ (ii) is known to hold whenever the tracial state space T (A) is a Bauer simplex”
Dimension, comparison, and almost finiteness (2017, background): “In fact, it is a conjecture of Toms and Winter that for infinite-dimensional simple separable unital nuclear C∗-algebras the following three conditions are equivalent: (i) finite nuclear dimension, (ii) Z-stability, (iii) strict comparison.”
Strict comparison and Z-absorption of nuclear C*-algebras (2011, background): “rption implies strict comparison for unital simple exact C∗-algebras [16]. W. Winter showed that any unital separable simple inﬁnite-dimensional C∗-algebra with ﬁnite nuclear dimension is Z-absorbing [25]. In the present paper we provide another partial answer to the conjecture above. Namely, it wil”
Tracially Z-absorbing C*-algebras (2012, result): “A property closely related to ours (the two coincide in certain cases) appeared as a technical step in Winter’s work concerning Z-absorption for C∗-algebras of finite nuclear dimension ([23, 25]) and the recent work of Matui-Sato on strict comparison and Z-absorption ([14]), and it may thus be profi”
Strict comparison and $$ \mathcal{Z} $$-absorption of nuclear C∗-algebras (2011, background): “We say that a unital C∗-algebra is Z-absorbing if A ∼= A ⊗ Z. H. Lin, Z. Niu and W. Winter proved that certain Z-absorbing C∗-algebras are classified by their ordered K-groups [13, 24].”
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].”