Strong pure infiniteness of crossed products. (English) Zbl 1398.46047

This paper provides sufficient criteria for the reduced crossed product of a separable \(C^*\)-algebra by a discrete exact group to be strongly purely infinite.
The notions of pure infiniteness and strong pure infiniteness for general \(C^*\)-algebras have been introduced in [E. Kirchberg and M. Rørdam, Am. J. Math. 122, No. 3, 637–666 (2000; Zbl 0968.46042); Adv. Math. 167, No. 2, 195–264 (2002; Zbl 1030.46075)]. The importance of strong pure infiniteness is due to a remarkable result of the first author: nuclear, separable, stable, strongly purely infinite \(C^*\)-algebras are classified by a \(KK\)-theoretic invariant. The same result is not known if one relaxes strong pure infiniteness to pure infiniteness. On the other hand, it remains an open question whether the notions of pure infiniteness and strong pure infiniteness agree (they do for simple \(C^*\)-algebras).
Thus, the results of the paper show that certain (non-simple) \(C^*\)-algebras arising as reduced crossed products are strongly purely infinite, which makes them accessible to \(KK\)-theoretic classification. Further, the paper can be understood as a first step towards determining if reduced crossed products are possible candidates that distinguish the notions of strong pure infiniteness and pure infiniteness.


46L35 Classifications of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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