×
Gabe, James

Traceless AF embeddings and unsuspended \(E\)-theory. (English) Zbl 1466.46050

Geom. Funct. Anal. 30, No. 2, 323-333 (2020).
The main result of the paper is:
{Theorem A.} For a separable exact \(C^*\)-algebra \(A\), the following conditions are equivalent:
●
\(A\) embeds into Rørdam’s purely infinite ASH algebra \(A_{[0,1]}\) (which is an inductive limit of cones over matrix algebras);
●
\(A\) embeds into the cone over the Cuntz algebra \(\mathcal O_2\);
●
\(A\) embeds into a zero-homotopic \(C^*\)-algebra;
●
the primitive ideal space \(\operatorname{Prim}A\) has no non-empty compact open subsets.
This result has a series of interesting corollaries. It is shown that if \(A\) satisfies the conditions of Theorem A, then it is AF embeddable (and, in particular, quasidiagonal). If \(A\) is additionally traceless, then the conditions of Theorem A are equivalent to AF embeddability, to quasidiagonality and to stable finiteness. If \(A\) is a nuclear \(C^*\)-algebra, then the conditions of Theorem A are equivalent to unsuspendability of \(E\)-theory (\(E(A,B)\cong [[A,B]]\) for any separable stable \(C^*\)-algebra \(B\), where \([[\cdot,\cdot]]\) denotes the homotopy classes of asymptotic homomorphisms.
Reviewer: Vladimir M. Manuilov (Moskva)

Cited in 6 Documents

MSC:

46L35 Classifications of \(C^*\)-algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L05 General theory of \(C^*\)-algebras
46M40 Inductive and projective limits in functional analysis
47A66 Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators
46L85 Noncommutative topology

Keywords:

AF embeddability; traceless \(\text C^*\)-algebras; unsuspended \(E\)-theory
Cite Review PDF
Full Text: DOI arXiv

References:

[1] Blackadar, B.; Kirchberg, E., Generalized inductive limits of finite-dimensional \({\rm C}^{\ast }\)-algebras, Math. Ann., 307, 3, 343-380 (1997) · Zbl 0874.46036 · doi:10.1007/s002080050039
[2] Blanchard, E., Subtriviality of continuous fields of nuclear \({\rm C}^{\ast }\)-algebras, J. Reine Angew. Math., 489, 133-149 (1997) · Zbl 0885.46047
[3] Brown, N., AF embeddability of crossed products of AF algebras by the integers, J. Funct. Anal., 160, 1, 150-175 (1998) · Zbl 0931.46043 · doi:10.1006/jfan.1998.3339
[4] A. Connes. Classification of injective factors. Cases \(II_1,~II_{\infty },~III_{\lambda },~\lambda \ne 1\). Ann. of Math. (2), (1)104 (1976), 73-115. · Zbl 0343.46042
[5] Connes, A.; Higson, N., Déformations, morphismes asymptotiques et \(K\)-théorie bivariante, C. R. Acad. Sci. Paris Sér. I Math, 311, 2, 101-106 (1990) · Zbl 0717.46062
[6] Dadarlat, M.; Loring, TA, \(K\)-homology, asymptotic representations, and unsuspended \(E\)-theory, J. Funct. Anal., 126, 2, 367-383 (1994) · Zbl 0863.46045 · doi:10.1006/jfan.1994.1151
[7] Dadarlat, M.; Pennig, U., Connective \({\rm C}^{\ast }\)-algebras, J. Funct. Anal., 272, 12, 4919-4943 (2017) · Zbl 1377.46038 · doi:10.1016/j.jfa.2017.02.009
[8] Dadarlat, M.; Pennig, U., Deformations of nilpotent groups and homotopy symmetric \({\rm C}^{\ast }\)-algebras, Math. Ann., 367, 1-2, 121-134 (2017) · Zbl 1359.46063 · doi:10.1007/s00208-016-1379-0
[9] J. Gabe. A new proof of Kirchberg’s \({\cal{O}}_2\)-stable classification. To appear in J. Reine Angew. Math. (arXiv:1706.03690v2) · Zbl 1461.46059
[10] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott. Continuous lattices and domains, volume 93 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (2003). · Zbl 1088.06001
[11] E. Kirchberg. The classification of purely infinite \({\rm C}^{\ast }\)-algebras using Kasparov’s theory. (1994).
[12] E. Kirchberg. Exact \({\rm C}^{\ast }\)-algebras, tensor products, and the classification of purely infinite algebras. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 943-954. Birkhäuser, Basel (1995). · Zbl 0897.46057
[13] E. Kirchberg. Das nicht-kommutative Michael-Auswahlprinzip und die Klassifikation nicht-einfacher Algebren. \(In {\rm C}^{\ast }\)-algebras (Münster, 1999), pages 92-141. Springer, Berlin (2000). · Zbl 0976.46051
[14] Kirchberg, E.; Phillips, NC, Embedding of exact \({\rm C}^{\ast }\)-algebras in the Cuntz algebra \({\cal{O}}_2 \), J. Reine Angew. Math., 525, 17-53 (2000) · Zbl 0973.46048 · doi:10.1515/crll.2000.065
[15] Kirchberg, E.; Rørdam, M., Non-simple purely infinite \({\rm C}^{\ast }\)-algebras, Amer. J. Math., 122, 3, 637-666 (2000) · Zbl 0968.46042 · doi:10.1353/ajm.2000.0021
[16] Kirchberg, E.; Rørdam, M., Infinite non-simple \({\rm C}^{\ast }\)-algebras: absorbing the Cuntz algebras \(\cal{O}_{\infty } \), Adv. Math., 167, 2, 195-264 (2002) · Zbl 1030.46075 · doi:10.1006/aima.2001.2041
[17] Kirchberg, E.; Rørdam, M., Purely infinite \({\rm C}^{\ast }\)-algebras: ideal-preserving zero homotopies, Geom. Funct. Anal., 15, 2, 377-415 (2005) · Zbl 1092.46044 · doi:10.1007/s00039-005-0510-2
[18] Ozawa, N., Homotopy invariance of AF-embeddability, Geom. Funct. Anal., 13, 1, 216-222 (2003) · Zbl 1028.46090 · doi:10.1007/s000390300005
[19] Ozawa, N.; Rørdam, M.; Sato, Y., Elementary amenable groups are quasidiagonal, Geom. Funct. Anal., 25, 1, 307-316 (2015) · Zbl 1335.46053 · doi:10.1007/s00039-015-0315-x
[20] Pasnicu, C.; Rørdam, M., Purely infinite \({\rm C}^{\ast }\)-algebras of real rank zero, J. Reine Angew. Math., 613, 51-73 (2007) · Zbl 1162.46029
[21] G. K. Pedersen. \({\rm C}^{\ast }\)-algebras and their automorphism groups, volume 14 of London Mathematical Society Monographs. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1979). · Zbl 0416.46043
[22] N. C. Phillips. A classification theorem for nuclear purely infinite simple \({\rm C}^{\ast }\)-algebras. Doc. Math., 5 (2000), 49-114 (electronic). · Zbl 0943.46037
[23] Pimsner, M., Embedding some transformation group \({\rm C}^{\ast } \)-algebras into AF-algebras, Ergodic Theory Dynam. Systems, 3, 4, 613-626 (1983) · Zbl 0582.46063 · doi:10.1017/S0143385700002182
[24] Pimsner, M.; Voiculescu, D., Imbedding the irrational rotation \(\text{C}^{\ast } \)-algebra into an AF-algebra, J. Operator Theory, 4, 1, 201-210 (1980) · Zbl 0525.46031
[25] Rørdam, M., A purely infinite AH-algebra and an application to AF-embeddability, Israel J. Math., 141, 61-82 (2004) · Zbl 1062.46051 · doi:10.1007/BF02772211
[26] Rørdam, M., The stable and the real rank of \({\cal{Z}} \)-absorbing \({\rm C}^{\ast } \)-algebras, Internat. J. Math., 15, 10, 1065-1084 (2004) · Zbl 1077.46054 · doi:10.1142/S0129167X04002661
[27] Tikuisis, A.; White, S.; Winter, W., Quasidiagonality of nuclear \({\rm C}^{\ast }\)-algebras, Ann. of Math. (2), 185, 1, 229-284 (2017) · Zbl 1367.46044 · doi:10.4007/annals.2017.185.1.4
[28] Voiculescu, D., A note on quasi-diagonal \({\rm C}^{\ast }\)-algebras and homotopy, Duke Math. J., 62, 2, 267-271 (1991) · Zbl 0833.46055 · doi:10.1215/S0012-7094-91-06211-3
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.