[Paper Review] On the KK-theory of strongly self-absorbing C*-algebras

Marius Dădărlat, Wilhelm Winter|ArXiv.org|Apr 4, 2007

Advanced Operator Algebra Research5 references3 citations

TL;DR

This paper establishes that for $K_1$-injective, strongly self-absorbing $C^*$-algebras $\mathcal{D}$, any two unital $*$-homomorphisms $\sigma, \gamma: \mathcal{D} \to A \otimes \mathcal{D}$ are asymptotically unitarily equivalent, not just approximately. This implies that all unital endomorphisms of $\mathcal{D}$ are asymptotically inner and that the spaces of automorphisms and endomorphisms of $\mathcal{D}$ are compactly contractible. The result provides a complete description of $KK(\mathcal{D}, A \otimes \mathcal{D})$ in terms of $*$-homomorphisms and asymptotic unitary equivalence.

ABSTRACT

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 $K_0(A\ot \Dh)$.

Motivation & Objective

• To establish asymptotic unitary equivalence of unital $*$-homomorphisms from a strongly self-absorbing $C^*$-algebra $\mathcal{D}$ to $A \otimes \mathcal{D}$, under $K_1$-injectivity.
• To characterize the Kasparov group $KK(\mathcal{D}, A \otimes \mathcal{D})$ using $*$-homomorphisms and asymptotic unitary equivalence.
• To show that the space of automorphisms and unital endomorphisms of $\mathcal{D}$ is compactly contractible in the point-norm topology.
• To prove $KK_i(\mathcal{D}, \mathcal{D} \otimes A) \cong K_i(\mathcal{D} \otimes A)$ for $i = 0,1$ without assuming the UCT.
• To provide $K$-theoretic characterizations of $\mathcal{O}_2$ and the universal UHF algebra $\mathcal{Q}$ without UCT assumptions.

Proposed method

• Leverages the known approximate unitary equivalence of unital $*$-homomorphisms $\sigma, \gamma: \mathcal{D} \to A \otimes \mathcal{D}$ for strongly self-absorbing $\mathcal{D}$, and strengthens it to asymptotic unitary equivalence under $K_1$-injectivity.
• Uses continuous paths of unitaries in $A \otimes \mathcal{D}$ to implement the intertwining of $\sigma$ and $\gamma$, with the path starting at the identity.
• Applies results from continuous fields of $C^*$-algebras and the theory of asymptotic unitary equivalence to deduce homotopy triviality of $\mathrm{Aut}(\mathcal{D})$ and $\mathrm{End}(\mathcal{D})$.
• Establishes isomorphism between $KK(\mathcal{D}, A \otimes \mathcal{D})$ and $K_0(A \otimes \mathcal{D})$ via a decomposition of $KK$-classes into differences of $*$-homomorphisms.
• Employs the universal property of strongly self-absorbing algebras and asymptotic multiplicativity of u.c.p. maps to prove $\mathcal{D} \cong \mathcal{D} \otimes \mathcal{Q}$ under $K_0$-divisibility and quasidiagonality.
• Uses the Künneth formula and $K$-theory conditions to characterize $\mathcal{Q}$ and $\mathcal{O}_2$ via $K_0$ and $K_1$-theoretic data without UCT.

Experimental results

Research questions

• RQ1 Are two unital $*$-homomorphisms $\sigma, \gamma: \mathcal{D} \to A \otimes \mathcal{D}$ asymptotically unitarily equivalent when $\mathcal{D}$ is strongly self-absorbing and $K_1$-injective?
• RQ2 Can the Kasparov group $KK(\mathcal{D}, A \otimes \mathcal{D})$ be fully described using $*$-homomorphisms and asymptotic unitary equivalence?
• RQ3 Is the space of automorphisms of a $K_1$-injective strongly self-absorbing $\mathcal{D}$ compactly contractible in the point-norm topology?
• RQ4 Does $KK_i(\mathcal{D}, \mathcal{D} \otimes A)$ isomorphically identify with $K_i(\mathcal{D} \otimes A)$ for $i = 0,1$ without assuming the UCT?
• RQ5 Can $\mathcal{O}_2$ and the universal UHF algebra $\mathcal{Q}$ be characterized purely by $K$-theoretic conditions, independent of the UCT?

Key findings

• Any two unital $*$-homomorphisms $\sigma, \gamma: \mathcal{D} \to A \otimes \mathcal{D}$ are asymptotically unitarily equivalent if $\mathcal{D}$ is strongly self-absorbing and $K_1$-injective.
• All unital endomorphisms of $\mathcal{D}$ are asymptotically inner, meaning they are implemented by a continuous path of unitaries starting at the identity.
• The space of automorphisms of $\mathcal{D}$ is compactly contractible: $[X, \mathrm{Aut}(\mathcal{D})]$ is trivial for any compact Hausdorff space $X$, in the point-norm topology.
• The Kasparov group $KK(\mathcal{D}, A \otimes \mathcal{D})$ is isomorphic to $K_0(A \otimes \mathcal{D})$, with elements represented as $[\varphi] - n[\iota]$ for $*$-homomorphisms $\varphi$ and inclusion $\iota$.
• Two $*$-homomorphisms $\varphi, \psi: \mathcal{D} \to \mathcal{K} \otimes A \otimes \mathcal{D}$ with $\varphi(1) = \psi(1) = e$ represent the same $KK$-class if and only if they are asymptotically unitarily equivalent via a path of unitaries in $e(\mathcal{K} \otimes A \otimes \mathcal{D})e$ starting at $e$. $KK_i(\mathcal{D}, \mathcal{D} \otimes A) \cong K_i(\mathcal{D} \otimes A)$ for $i = 0,1$, without UCT assumptions.
• The universal UHF algebra $\mathcal{Q}$ is characterized as the unique separable, unital, strongly self-absorbing, quasidiagonal $C^*$-algebra with $K_0(\mathcal{Q}) \cong \mathbb{Q}$ and $K_1(\mathcal{Q}) \otimes \mathbb{Q} = 0$, without relying on the UCT.