## 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.

### MSC:

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

### Citations:

Zbl 0968.46042; Zbl 1030.46075
Full Text:

### References:

 [1] Archbold, R. J. and Spielberg, J. S.. Topologically free actions and ideals in discrete C*-dynamical systems. Proc. Edinb. Math. Soc. (2)37 (1994), 119-124. doi:10.1017/S0013091500018733 · Zbl 0799.46076 [2] Arzhantseva, G., Guba, V. and Sapir, M.. Metrics on diagram groups and uniform embeddings in Hilbert space. Comment. Math. Helv.81 (2006), 911-929. doi:10.4171/CMH/80 · Zbl 1166.20031 [3] Blanchard, E. and Kirchberg, E.. Non-simple purely infinite C*-algebras: the Hausdorff case. J. Funct. Anal.207 (2004), 461-513. doi:10.1016/j.jfa.2003.06.008 · Zbl 1048.46049 [4] Brown, L. G. and Pedersen, G. K.. C*-algebras of real rank zero. J. Funct. Anal.99 (1991), 131-149. doi:10.1016/0022-1236(91)90056-B · Zbl 0776.46026 [5] Cuntz, J., K-theory for certain C*-algebras, Ann. of Math. (2), 113, 181-197, (1981) · Zbl 0437.46060 [6] Elliott, G. A., Some simple C*-algebras constructed as crossed products with discrete outer automorphism groups, Publ. Res. Inst. Math. Sci., 16, 299-311, (1980) · Zbl 0438.46044 [7] Exel, R., Laca, M. and Quigg, J.. Partial dynamical systems and C*-algebras generated by partial isometries. J. Operator Theory47 (2002), 169-186. · Zbl 1029.46106 [8] Haagerup, S., Haagerup, U. and Ramirez-Solano, M.. A computational approach to the Thompson group F. Int. J. Algebra Comput.25 (2015), 875-885. doi:10.1142/S0218196715500022 · Zbl 1335.20042 [9] Haagerup, U. and Picioroaga, G.. New presentations of Thompson’s groups and applications. J. Operator Theory66 (2011), 217-232. · Zbl 1249.22005 [10] Jolissaint, P. and Robertson, G.. Simple purely infinite C*-algebras and n-filling actions. J. Funct. Anal.175 (2000), 197-213. doi:10.1006/jfan.2000.3608 · Zbl 0993.46033 [11] Kawamura, S. and Tomiyama, J.. Properties of topological dynamical systems and corresponding C*-algebras. Tokyo J. Math.13 (1990), 251-257. doi:10.3836/tjm/1270132260 · Zbl 0724.54037 [12] Kirchberg, E. and Rørdam, M.. Non-simple purely infinite C*-algebras. Amer. J. Math.122 (2000), 637-666. doi:10.1353/ajm.2000.0021 · Zbl 0968.46042 [13] Kirchberg, E. and Rørdam, M.. Infinite non-simple C*-algebras: absorbing the Cuntz algebras 𝓞_{∞}. Adv. Math.167(2) (2002), 195-264. doi:10.1006/aima.2001.2041 · Zbl 1030.46075 [14] Kirchberg, E. and Sierakowski, A.. Filling families and strong pure infiniteness. Preprint, 2015, arXiv:1503.08519v2. [15] Kirchberg, E. and Wassermann, S.. Exact groups and continuous bundles of C*-algebras. Math. Ann.315 (1999), 169-203. doi:10.1007/s002080050364 · Zbl 0946.46054 [16] Kishimoto, A., Outer automorphisms and reduced crossed products of simple C*-algebras, Comm. Math. Phys., 81, 429-435, (1981) · Zbl 0467.46050 [17] Laca, M. and Spielberg, J.. Purely infinite C*-algebras from boundary actions of discrete groups. J. reine angew. Math.480 (1996), 125-139. · Zbl 0863.46044 [18] Olesen, D. and Pedersen, G. K.. Applications of the Connes spectrum to C*-dynamical systems. III. J. Funct. Anal.45 (1982), 357-390. doi:10.1016/0022-1236(82)90011-8 · Zbl 0511.46064 [19] Pedersen, G. K., C*-algebras and Their Automorphism Groups, (1979), Academic Press: Academic Press, London · Zbl 0416.46043 [20] Renault, J., The ideal structure of groupoid crossed product C*-algebras, with an appendix by Georges Skandalis, J. Operator Theory, 25, 3-36, (1991) [21] Rørdam, M., Larsen, F. and Laustsen, N.. An Introduction to K-theory for C*-algebras. Cambridge University Press, Cambridge, 2000. · Zbl 0967.19001 [22] Sierakowski, A., The ideal structure of reduced crossed products, Münster J. Math., 3, 223-248, (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.