On the Independence of K-Theory and Stable Rank for Simple C*-Algebras

Abstract

Jiang and Su and (independently) Elliott discovered a simple, nuclear, infinite-dimensional C * -algebra Z having the same Elliott invariant as the complex numbers. For a nuclear C * -algebra A with weakly unperforated K * -group the Elliott invariant of A ⊗ Z is isomorphic to that of A. Thus, any simple nuclear C * -algebra A having a weakly unperforated K * -group which does not absorb Z provides a counterexample to Elliott's conjecture that the simple nuclear C * -algebras will be classified by the Elliott invariant. In the sequel we exhibit a separable, infinite-dimensional, stably finite instance of such a non-Z-absorbing algebra A, and so provide a counterexample to the Elliott conjecture for the class of simple, nuclear, infinite-dimensional, stably finite, separable C * -algebras.

K-theoretic rigidity and slow dimension growth

“…We now know that this conjecture, while true in considerable generality, is too much to hope for. The author showed in [24] and [25] that an additional condition-slow dimension growth-is required in general, a condition present in each of the examples listed above. Finally, one needs simplicity in order to avoid phenomena detectable only using K-theory with (mod p)-coefficients (see [7] and [8]).…”

Regularity properties in the classification program for separable amenable C*-algebras

“…Remarkably, there exist algebras satisfying the hypotheses of the above theorem which are not Z-stable ( [51], [56], [57] We shall see later that for a substantial class of simple, separable, amenable, and stably finite C * -algebras, all three of our regularity properties are equivalent. Moreover, the algebras in this class which do satisfy these three properties also satisfy (EC).…”

Z-stability and infinite tensor powers of C∗-algebras

“…By the theorem of Kirchberg cited in (a) we have that A ⊗ A is purely infinite, and so following the arguments of (a) we see that A satisfies the hypotheses of Theorem 1.1. Other examples were produced by the second named author in [14] and [12]. These algebras are ASH and non-type-I, and so satisfy the hypotheses of Theorem 1.1 by the arguments of (b) above.…”

“…(c) properly infinite C * -algebras; (d) real rank zero C * -algebras without characters; (e) C * -algebras arising from minimal dynamics on a compact infinite Hausdorff space; (f) algebras considered pathological with respect to the strong form of Elliott's classification conjecture for separable amenable C * -algebras [10,12,14].…”