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