On the KK-theory of strongly self-absorbing C*-algebras

By Marius Dadarlat, W. Winter. 2007

48 citations recorded by the source corpus.

Let $\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\Dh$ be strongly self-absorbing. It is known that any two unital $^*$-homomorphisms from $\Dh$ to $A \otimes \Dh$ are approximately unitarily equivalent. We show that, if $\Dh$ is also $K_{1}$-injective, they are even asymptotically unitarily equivalent. This in particular implies that any unital endomorphism of $\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\Dh$ is compactly-contractible (in the point-norm topology) in the sense that for any compact Hausdorff space $X$, the set of homotopy classes $[X,\Aut(\Dh)]$ reduces to a point. The respective statement holds for the space of unital endomorphisms of $\Dh$. As an application, we give a description of the Kasparov group $KK(\Dh, A\ot \Dh)$ in terms of $^*$-homomorphisms and asymptotic unitary equivalence. Along the way, we show that the Kasparov group $KK(\Dh, A\ot \Dh)$ is isomorphic to $K_0(A\ot \Dh)$.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/60b663275c8535507092284c43ccb3f474035142

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In Algebra and Number Theory

1. G'abor Szab'o
2. W. Winter
3. Ilan Hirshberg
4. Marius Dadarlat
5. M. Rørdam
6. U. Pennig
7. David Kerr
8. Yasuhiko Sato

In Mathematical Physics

1. G'abor Szab'o
2. W. Winter
3. Ilan Hirshberg
4. Marius Dadarlat
5. M. Rørdam
6. David Kerr
7. U. Pennig
8. Yasuhiko Sato

Citations in the corpus

• Localizing the Elliott conjecture at strongly self-absorbing C*-algebras (2007, background): “…generalized dimension drop intervals and show how such an isomorphism may be used to obtain an asymptotic unitary intertwining between tensor products with Z. 4.1 Notation: For 0 m ≤ n ∈ N and unital C∗-algebrasA1, . . . , Am, B1, . . . , Bn let i1, . . . , im ∈ {1, . . . , n be pairwise…”
• Continuous fields of C*-algebras over finite dimensional spaces (2006, background): “By a result of [16], every unital ∗-endomorphism of D is a KK-equivalence.”
• $${\mathcal{C}}_{0}$$ (X)-algebras, stability and strongly self-absorbing $${\mathcal{C}}^{*}$$ -algebras (2007, background): “In [ 9 ], Duaduarlat and the third named author derive the respective result for an arbitrary K1-injective strongly self-absorbing C ∗-algebra.”
• A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group (2014, background): “Since both U(D ⊗ Mn(C)) and Aut(D) are path connected by [17], [15] and respectively [6] we conclude that Aut(D⊗Mn(C)) is path-connected as well.”
• Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions (2016, background): “Using again that D is strongly self-absorbing, the flip automorphism on D ⊗ D is homotopic to the identity map; see [13].”
• Rokhlin actions and self-absorbing C*-algebras (2005, background): “Related results have been obtained in [Hirshberg, Rørdam and Winter 2007], [Hirshberg and Winter 2007], and [Dadarlat and Winter 2007b].”