Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nuclear dimension and Z-stability of pure C*-algebras   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.

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The categories of this paper are: atom-ph
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-ph