Strongly self-absorbing 𝐶*-algebras

Toms, Andrew; Winter, Wilhelm

doi:10.1090/S0002-9947-07-04173-6

Strongly self-absorbing $C^{*}$-algebras
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by Andrew S. Toms and Wilhelm Winter PDF

Trans. Amer. Math. Soc. 359 (2007), 3999-4029 Request permission

Abstract:

Say that a separable, unital $C^*$-algebra $\mathcal {D} \ncong \mathbb {C}$ is strongly self-absorbing if there exists an isomorphism $\varphi : \mathcal {D} \to \mathcal {D} \otimes \mathcal {D}$ such that $\varphi$ and $\mathrm {id}_{\mathcal {D}} \otimes \mathbf {1}_{\mathcal {D}}$ are approximately unitarily equivalent $*$-homomorphisms. We study this class of algebras, which includes the Cuntz algebras $\mathcal {O}_2$, $\mathcal {O}_{\infty }$, the UHF algebras of infinite type, the Jiang–Su algebra $\mathcal {Z}$ and tensor products of $\mathcal {O}_{\infty }$ with UHF algebras of infinite type. Given a strongly self-absorbing $C^{*}$-algebra $\mathcal {D}$ we characterise when a separable $C^*$-algebra absorbs $\mathcal {D}$ tensorially (i.e., is $\mathcal {D}$-stable), and prove closure properties for the class of separable $\mathcal {D}$-stable $C^*$-algebras. Finally, we compute the possible $K$-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing $C^*$-algebras.

References

Brown, N.: Invariant means and finite representation theory of $C^{*}$-algebras, preprint math.OA/0304009v1 (2003).
Joachim Cuntz, Simple $C^*$-algebras generated by isometries, Comm. Math. Phys. 57 (1977), no. 2, 173–185. MR 467330
Joachim Cuntz, $K$-theory for certain $C^{\ast }$-algebras, Ann. of Math. (2) 113 (1981), no. 1, 181–197. MR 604046, DOI 10.2307/1971137
Marius Dădărlat, Reduction to dimension three of local spectra of real rank zero $C^*$-algebras, J. Reine Angew. Math. 460 (1995), 189–212. MR 1316577, DOI 10.1515/crll.1995.460.189
Edward G. Effros and Jonathan Rosenberg, $C^{\ast }$-algebras with approximately inner flip, Pacific J. Math. 77 (1978), no. 2, 417–443. MR 510932
George A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. MR 397420, DOI 10.1016/0021-8693(76)90242-8
George A. Elliott, The classification problem for amenable $C^*$-algebras, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 922–932. MR 1403992
George A. Elliott, On the classification of $C^*$-algebras of real rank zero, J. Reine Angew. Math. 443 (1993), 179–219. MR 1241132, DOI 10.1515/crll.1993.443.179
George A. Elliott, Guihua Gong, and Liangqing Li, Approximate divisibility of simple inductive limit $C^\ast$-algebras, Operator algebras and operator theory (Shanghai, 1997) Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 87–97. MR 1667656, DOI 10.1090/conm/228/03283
Elliott, G. A., Gong, G. and Li, L.: On the classification of simple inductive limit $C^*$-algebras, II: The isomorphism theorem, preprint.
Elliott, G. A. and Rørdam, M.: Perturbation of Hausdorff moment sequences, and an application to the theory of $C^{*}$-algebras of real rank zero, preprint math.OA/0502239v1 (2005).
Guihua Gong, On the classification of simple inductive limit $C^*$-algebras. I. The reduction theorem, Doc. Math. 7 (2002), 255–461. MR 2014489
Guihua Gong, Xinhui Jiang, and Hongbing Su, Obstructions to $\scr Z$-stability for unital simple $C^*$-algebras, Canad. Math. Bull. 43 (2000), no. 4, 418–426. MR 1793944, DOI 10.4153/CMB-2000-050-1
Hirshberg, I. and Winter, W.: Rokhlin actions and self-absorbing $C^{*}$-algebras, preprint math.OA/0505356 (2005).
Ivanescu, C.: On the classification of simple approximately subhomogeneous $C^{*}$-algebras not necessarily of real rank zero, preprint math.OA/0312044v2 (2003).
Jiang, X.: Nonstable $K$-theory for $\mathcal {Z}$-stable $C^{*}$-algebras, preprint math.OA/9707228 (1997).
Xinhui Jiang and Hongbing Su, On a simple unital projectionless $C^*$-algebra, Amer. J. Math. 121 (1999), no. 2, 359–413. MR 1680321
Xinhui Jiang and Hongbing Su, A classification of simple limits of splitting interval algebras, J. Funct. Anal. 151 (1997), no. 1, 50–76. MR 1487770, DOI 10.1006/jfan.1997.3120
Kirchberg, E.: The classification of purely infinite $C^*$-algebras using Kasparov’s theory, Fields Institute Comm., to appear.
Kirchberg, E.: Permanence properties of strongly purely infinite $C^{*}$-algebras, preprint (2004).
Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
Eberhard Kirchberg and Wilhelm Winter, Covering dimension and quasidiagonality, Internat. J. Math. 15 (2004), no. 1, 63–85. MR 2039212, DOI 10.1142/S0129167X04002119
Huaxin Lin, Classification of simple tracially AF $C^*$-algebras, Canad. J. Math. 53 (2001), no. 1, 161–194. MR 1814969, DOI 10.4153/CJM-2001-007-8
Jesper Mygind, Classification of certain simple $C^*$-algebras with torsion in $K_1$, Canad. J. Math. 53 (2001), no. 6, 1223–1308. MR 1863849, DOI 10.4153/CJM-2001-046-2
Gert K. Pedersen, $C^{\ast }$-algebras and their automorphism groups, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR 548006
Francesc Perera and Mikael Rørdam, AF-embeddings into $C^*$-algebras of real rank zero, J. Funct. Anal. 217 (2004), no. 1, 142–170. MR 2097610, DOI 10.1016/j.jfa.2004.05.001
Eberhard Kirchberg and N. Christopher Phillips, Embedding of exact $C^*$-algebras in the Cuntz algebra $\scr O_2$, J. Reine Angew. Math. 525 (2000), 17–53. MR 1780426, DOI 10.1515/crll.2000.065
Phillips, N. C.: Cancellation and stable rank for direct limits of recursive subhomogeneous algebras, preprint math.OA/0101157v1 (2001).
Phillips, N. C.: Recursive subhomogeneous algebras, Trans. Amer. Math. Soc., to appear.
M. Rørdam and E. Størmer, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia of Mathematical Sciences, vol. 126, Springer-Verlag, Berlin, 2002. Operator Algebras and Non-commutative Geometry, 7. MR 1878881, DOI 10.1007/978-3-662-04825-2
Mikael Rørdam, A simple $C^*$-algebra with a finite and an infinite projection, Acta Math. 191 (2003), no. 1, 109–142. MR 2020420, DOI 10.1007/BF02392697
Mikael Rørdam, The stable and the real rank of $\scr Z$-absorbing $C^*$-algebras, Internat. J. Math. 15 (2004), no. 10, 1065–1084. MR 2106263, DOI 10.1142/S0129167X04002661
Irina Stevens, Simple approximate circle algebras, Operator algebras and their applications, II (Waterloo, ON, 1994/1995) Fields Inst. Commun., vol. 20, Amer. Math. Soc., Providence, RI, 1998, pp. 97–104. MR 1643180, DOI 10.1046/j.1440-1754.1998.00164.x
Masamichi Takesaki, Theory of operator algebras. I, Springer-Verlag, New York-Heidelberg, 1979. MR 548728
Klaus Thomsen, Limits of certain subhomogeneous $C^*$-algebras, Mém. Soc. Math. Fr. (N.S.) 71 (1997), vi+125 pp. (1998) (English, with English and French summaries). MR 1649315, DOI 10.24033/msmf.385
Andrew Toms, On the independence of $K$-theory and stable rank for simple $C^*$-algebras, J. Reine Angew. Math. 578 (2005), 185–199. MR 2113894, DOI 10.1515/crll.2005.2005.578.185
Toms, A. S. and Winter, W.: $\mathcal {Z}$-stable ASH algebras, preprint math.OA/0508218 (2005), to appear in Can. J. Math.
Jesper Villadsen, Simple $C^*$-algebras with perforation, J. Funct. Anal. 154 (1998), no. 1, 110–116. MR 1616504, DOI 10.1006/jfan.1997.3168
Jesper Villadsen, On the stable rank of simple $C^\ast$-algebras, J. Amer. Math. Soc. 12 (1999), no. 4, 1091–1102. MR 1691013, DOI 10.1090/S0894-0347-99-00314-8
Wilhelm Winter, Covering dimension for nuclear $C^*$-algebras, J. Funct. Anal. 199 (2003), no. 2, 535–556. MR 1971906, DOI 10.1016/S0022-1236(02)00109-X
Wilhelm Winter, Decomposition rank of subhomogeneous $C^*$-algebras, Proc. London Math. Soc. (3) 89 (2004), no. 2, 427–456. MR 2078703, DOI 10.1112/S0024611504014716
Wilhelm Winter, On topologically finite-dimensional simple $C^\ast$-algebras, Math. Ann. 332 (2005), no. 4, 843–878. MR 2179780, DOI 10.1007/s00208-005-0657-z
Wilhelm Winter, On the classification of simple $\scr Z$-stable $C^*$-algebras with real rank zero and finite decomposition rank, J. London Math. Soc. (2) 74 (2006), no. 1, 167–183. MR 2254559, DOI 10.1112/S0024610706022903
Winter, W.: Simple $C^{*}$-algebras with locally finite decomposition rank, preprint math.OA/0602617 (2006), to appear in J. Funct. Anal.