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Traceless AF embeddings and unsuspended \(\varvec{E}\)-theory

Abstract

I show that quasidiagonality and AF embeddability are equivalent properties for traceless \(\mathrm C^*\)-algebras and are characterised in terms of the primitive ideal space. For nuclear \(\mathrm C^*\)-algebras the same characterisation determines when Connes and Higson’s E-theory can be unsuspended.

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Notes

  1. Recall that a not necessarily unital \(\mathrm C^*\)-algebra A is stably finite if for every \(n\in {\mathbb {N}}\), \(M_n({\widetilde{A}})\) has no non-unitary isometries. In particular, stably projectionless \(\mathrm C^*\)-algebras are stably finite.

  2. One does not need to know how the Lawson topology is defined, nor what having small semilattices means, in order to understand the proof.

  3. The theorem is applicable for domains and thus also for continuous lattices since a continuous lattice is a domain which is also a complete lattice, see [GHK+03, Defintion I-1.6].

  4. Arc chains are defined as subsets of pospaces. By [GHK+03, Proposition VI-1.14] it follows that any compact semilattice is a pospace, so this makes sense in our case.

  5. This is applicable since the order on \({\mathcal {I}}(A)\) is semiclosed with respect to the Lawson topology due to the remark following [GHK+03, Definition III-5.1].

  6. In fact, if \(0< I < A\) in \({\mathcal {C}}\), then [0, I) and (IA] are both clopen subsets of \({\mathcal {C}}{\setminus }\{I\}\) in the order topology, so any such I is a cut point. The elements 0 and A are not cut-points as these are the minimal and maximal element respectively in \({\mathcal {C}}\).

  7. Alternative proof: The map \(\Phi :[0,1] \rightarrow {\mathcal {I}}(C_0((0,1],{\mathcal {O}}_2))\) given by \(\Phi (t) = C_0((\tfrac{1-t}{2}, \tfrac{1+t}{2}), {\mathcal {O}}_2)\) is a Cu-morphism for which \(\Phi ^{-1}(\{0\}) = \{ 0\}\). Hence \(\Phi \) induces an embedding \({\mathcal {A}}_{[0,1]} \hookrightarrow C_0((0,1],{\mathcal {O}}_2)\) by Theorems 4 and 5.

  8. In [DP17b] the same definition was called property (QH).

References

  • B. Blackadar and E. Kirchberg. Generalized inductive limits of finite-dimensional \({\rm C}^{\ast }\)-algebras. Math. Ann., (3)307 (1997), 343–380.

    MathSciNet  Article  Google Scholar 

  • E. Blanchard. Subtriviality of continuous fields of nuclear \({\rm C}^{\ast }\)-algebras. J. Reine Angew. Math., 489(1997), 133–149.

    MathSciNet  MATH  Google Scholar 

  • N. Brown. AF embeddability of crossed products of AF algebras by the integers. J. Funct. Anal., (1)160 (1998), 150–175.

    MathSciNet  Article  Google Scholar 

  • 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.

  • A. Connes and N. Higson. Déformations, morphismes asymptotiques et \(K\)-théorie bivariante. C. R. Acad. Sci. Paris Sér. I Math., (2)311 (1990), 101–106.

    MathSciNet  MATH  Google Scholar 

  • M. Dadarlat and T. A. Loring. \(K\)-homology, asymptotic representations, and unsuspended \(E\)-theory. J. Funct. Anal., (2)126 (1994), 367–383.

    MathSciNet  Article  Google Scholar 

  • M. Dadarlat and U. Pennig. Connective \({\rm C}^{\ast }\)-algebras. J. Funct. Anal., (12)272 (2017), 4919–4943.

    MathSciNet  Article  Google Scholar 

  • M. Dadarlat and U. Pennig. Deformations of nilpotent groups and homotopy symmetric \({\rm C}^{\ast }\)-algebras. Math. Ann., (1-2)367 (2017), 121–134.

    MathSciNet  Article  Google Scholar 

  • J. Gabe. A new proof of Kirchberg’s \({\cal{O}}_2\)-stable classification. To appear in J. Reine Angew. Math. (arXiv:1706.03690v2)

  • 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).

  • E. Kirchberg. The classification of purely infinite \({\rm C}^{\ast }\)-algebras using Kasparov’s theory. (1994).

  • 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).

  • 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).

  • E. Kirchberg and N. C. Phillips. Embedding of exact \({\rm C}^{\ast }\)-algebras in the Cuntz algebra \({\cal{O}}_{2}\). J. Reine Angew. Math., 525 (2000), 17–53.

    MathSciNet  Article  Google Scholar 

  • E. Kirchberg and M. Rørdam. Non-simple purely infinite \({\rm C}^{\ast }\)-algebras. Amer. J. Math., (3)122 (2000), 637–666.

    MathSciNet  Article  Google Scholar 

  • E. Kirchberg and M. Rørdam. Infinite non-simple \({\rm C}^{\ast }\)-algebras: absorbing the Cuntz algebras \({\cal{O}}_{\infty }\). Adv. Math., (2)167 (2002), 195–264.

    MathSciNet  Article  Google Scholar 

  • E. Kirchberg and M. Rørdam. Purely infinite \({\rm C}^{\ast }\)-algebras: ideal-preserving zero homotopies. Geom. Funct. Anal., (2)15 (2005), 377–415.

    MathSciNet  Article  Google Scholar 

  • N. Ozawa. Homotopy invariance of AF-embeddability. Geom. Funct. Anal., (1)13 (2003), 216–222.

    MathSciNet  Article  Google Scholar 

  • N. Ozawa, M. Rørdam, and Y. Sato. Elementary amenable groups are quasidiagonal. Geom. Funct. Anal., (1)25 (2015), 307–316.

    MathSciNet  Article  Google Scholar 

  • C. Pasnicu and M. Rørdam. Purely infinite \({\rm C}^{\ast }\)-algebras of real rank zero. J. Reine Angew. Math., 613 (2007), 51–73.

    MathSciNet  MATH  Google Scholar 

  • 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).

  • N. C. Phillips. A classification theorem for nuclear purely infinite simple \({\rm C}^{\ast }\)-algebras. Doc. Math., 5 (2000), 49–114 (electronic).

  • M. Pimsner. Embedding some transformation group \({\rm C}^{\ast } \)-algebras into AF-algebras. Ergodic Theory Dynam. Systems, (4)3 (1983), 613–626.

    MathSciNet  Article  Google Scholar 

  • M. Pimsner and D. Voiculescu. Imbedding the irrational rotation \(\text{ C }^{\ast }\)-algebra into an AF-algebra. J. Operator Theory, (1)4 (1980), 201–210.

    MathSciNet  MATH  Google Scholar 

  • M. Rørdam. A purely infinite AH-algebra and an application to AF-embeddability. Israel J. Math., 141 (2004), 61–82.

    MathSciNet  Article  Google Scholar 

  • M. Rørdam. The stable and the real rank of \({\cal{Z}}\)-absorbing \({\rm C}^{\ast }\)-algebras. Internat. J. Math., (10)15 (2004), 1065–1084.

    MathSciNet  Article  Google Scholar 

  • A. Tikuisis, S. White, and W. Winter. Quasidiagonality of nuclear \({\rm C}^{\ast }\)-algebras. Ann. of Math. (2), (1)185 (2017), 229–284.

    MathSciNet  Article  Google Scholar 

  • D. Voiculescu. A note on quasi-diagonal \({\rm C}^{\ast }\)-algebras and homotopy. Duke Math. J., (2)62 (1991), 267–271.

    MathSciNet  Article  Google Scholar 

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Acknowledgements

I would like to thank the referee for suggesting that I expand Theorem A. I am very grateful to Stuart White for several helpful comments and suggestions.

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Correspondence to James Gabe.

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This work was funded by the Carlsberg Foundation through an Internationalisation Fellowship.

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Gabe, J. Traceless AF embeddings and unsuspended \(\varvec{E}\)-theory. Geom. Funct. Anal. 30, 323–333 (2020). https://doi.org/10.1007/s00039-020-00528-2

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Keywords

  • AF embeddability
  • Traceless \(\mathrm C^*\)-algebras
  • Unsuspended E-theory

Mathematics Subject Classification

  • 46L05
  • 46L80