## 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.

This is a preview of subscription content, access via your institution.

## Notes

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

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].

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.

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].

In fact, if \(0< I < A\) in \({\mathcal {C}}\), then [0,

*I*) and (*I*,*A*] 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 \({\mathcal {C}}\).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.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.E. Blanchard. Subtriviality of continuous fields of nuclear \({\rm C}^{\ast }\)-algebras.

*J. Reine Angew. Math.*, 489(1997), 133–149.N. Brown. AF embeddability of crossed products of AF algebras by the integers.

*J. Funct. Anal.*, (1)160 (1998), 150–175.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.M. Dadarlat and T. A. Loring. \(K\)-homology, asymptotic representations, and unsuspended \(E\)-theory.

*J. Funct. Anal.*, (2)126 (1994), 367–383.M. Dadarlat and U. Pennig. Connective \({\rm C}^{\ast }\)-algebras.

*J. Funct. Anal.*, (12)272 (2017), 4919–4943.M. Dadarlat and U. Pennig. Deformations of nilpotent groups and homotopy symmetric \({\rm C}^{\ast }\)-algebras.

*Math. Ann.*, (1-2)367 (2017), 121–134.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.E. Kirchberg and M. Rørdam. Non-simple purely infinite \({\rm C}^{\ast }\)-algebras.

*Amer. J. Math.*, (3)122 (2000), 637–666.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.E. Kirchberg and M. Rørdam. Purely infinite \({\rm C}^{\ast }\)-algebras: ideal-preserving zero homotopies.

*Geom. Funct. Anal.*, (2)15 (2005), 377–415.N. Ozawa. Homotopy invariance of AF-embeddability.

*Geom. Funct. Anal.*, (1)13 (2003), 216–222.N. Ozawa, M. Rørdam, and Y. Sato. Elementary amenable groups are quasidiagonal.

*Geom. Funct. Anal.*, (1)25 (2015), 307–316.C. Pasnicu and M. Rørdam. Purely infinite \({\rm C}^{\ast }\)-algebras of real rank zero.

*J. Reine Angew. Math.*, 613 (2007), 51–73.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.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.M. Rørdam. A purely infinite AH-algebra and an application to AF-embeddability.

*Israel J. Math.*, 141 (2004), 61–82.M. Rørdam. The stable and the real rank of \({\cal{Z}}\)-absorbing \({\rm C}^{\ast }\)-algebras.

*Internat. J. Math.*, (10)15 (2004), 1065–1084.A. Tikuisis, S. White, and W. Winter. Quasidiagonality of nuclear \({\rm C}^{\ast }\)-algebras.

*Ann. of Math. (2)*, (1)185 (2017), 229–284.D. Voiculescu. A note on quasi-diagonal \({\rm C}^{\ast }\)-algebras and homotopy.

*Duke Math. J.*, (2)62 (1991), 267–271.

## 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.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

## Rights and permissions

## About this article

### Cite this article

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

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s00039-020-00528-2

### Keywords

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

### Mathematics Subject Classification

- 46L05
- 46L80