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# $\mathcal{Z}$-Stable ASH Algebras

Published:2008-06-01
Printed: Jun 2008
• Andrew S. Toms,
The Jiang--Su algebra $\mathcal{Z}$ has come to prominence in the classification program for nuclear $C^*$-algebras of late, due primarily to the fact that Elliott's classification conjecture in its strongest form predicts that all simple, separable, and nuclear $C^*$-algebras with unperforated $\mathrm{K}$-theory will absorb $\mathcal{Z}$ tensorially, i.e., will be $\mathcal{Z}$-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and $\mathcal{Z}$-stable $C^*$-algebras. We prove that virtually all classes of nuclear $C^*$-algebras for which the Elliott conjecture has been confirmed so far consist of $\mathcal{Z}$-stable $C^*$-algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible $C^*$-algebras are $\mathcal{Z}$-stable.
 Keywords: nuclear $C^*$-algebras, K-theory, classification
 MSC Classifications: 46L85 - Noncommutative topology [See also 58B32, 58B34, 58J22] 46L35 - Classifications of $C^*$-algebras