On the Classification of Simple Z‐Stable C*‐Algebras with Real Rank Zero and Finite Decomposition Rank

By W. Winter. 2005

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We show that, if A is a separable simple unital C*‐algebra that absorbs the Jiang–Su algebra Z tensorially and that has real rank zero and finite decomposition rank, then A is tracially approximately finite‐dimensional in the sense of Lin, without any restriction on the tracial state space. As a consequence, the Elliott conjecture is true for the class of C*‐algebras as above that, additionally, satisfy the universal coefficients theorem. In particular, such algebras are completely determined by their ordered K‐theory. They are approximately homogeneous of topological dimension less than or equal to three, approximately subhomogeneous of topological dimension at most two and their decomposition rank also is no greater than two.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bda5e5060071989fb5644d5b587b1b653a07c24d

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Citations in the corpus

On the KK-theory of strongly self-absorbing C*-algebras (2007, background): “[8], [10], [ 16 ], [17], [18] and [15] for already existing results in this direction.”
The Elliott conjecture for Villadsen algebras of the first type (2006, background): “These examples have legitimised the assumption of Z -stability in Elliott’s classification program, leading to the wide-ranging classification theorem of the second named author for ASH algebras of real rank zero [37,38].”
Approximate Homotopy of Homomorphisms from C(x) Into a Simple C*-Algebra (2006, background): “We refer the reader to [24], [28], [25], [27], [3], [45] and [46] for more information.”
$${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras (2007, background): “Absorption of the Jiang‐Su algebra seems to be particularly important, since it is automatic for the known classifiable algebras, whereas the known counterexamples to the Elliott conjecture fail to absorb Z. Algebras which absorb the Jiang‐Su algebra tend to be well-behaved in many respects, and thu”
Localizing the Elliott conjecture at strongly self-absorbing C*-algebras (2007, background): “1.4 Examples: The only known examples of strongly self-absorbing C∗-algebras are UHF algebras ([19]) of infinite type (i.e., every prime number that occurs in the respective supernatural number occurs with infinite multiplicity – in this case, we also say the supernatural number is of infinite…”