Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras

Abstract

Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${KK}_e(C,A{)}^{++}$ the set of those $\kappa$ for which $\kappa (K_0(C{)}_{+}\setminus \{0\})\subset K_0(A{)}_{+}\setminus \{0\}$ and $\kappa ([1_C])=[1_A]$. Suppose that $\kappa \in {KK}_e(C,A{)}^{++}.$ We show that there is a unital monomorphism $\varphi :C\to A$ such that $[\varphi ]=\kappa .$ Suppose that $C$ is a unital AH-algebra and $\lambda :\mathrm{T}(A)\to {\mathrm{T}}_{\mathtt{f}}(C)$ is a continuous affine map for which $\tau (\kappa ([p]))=\lambda (\tau )(p)$ for all projections $p$ in all matrix algebras of $C$ and any $\tau \in \mathrm{T}(A),$ where $\mathrm{T}(A)$ is the simplex of tracial states of $A$ and ${\mathrm{T}}_{\mathtt{f}}(C)$ is the convex set of faithful tracial states of $C.$ We prove that there is a unital monomorphism $\varphi :C\to A$ such that $\varphi$ induces both $\kappa$ and $\lambda .$ Suppose that $h:C\to A$ is a unital monomorphism and $γ\in \mathrm{Hom}(\Kone(C), \aff(A)).$ We show that there exists a unital monomorphism $\varphi :C\to A$ such that $[\varphi ]=[h]$ in $KK(C,A),$ $\tau \circ \varphi =\tau \circ h$ for all tracial states $\tau$ and the associated rotation map can be given by $\gamma .$ Applications to classification of simple C*-algebras are also given.

The new version made a correction and removed a number of typos

References in corpus (2)

• Localizing the Elliott conjecture at strongly self-absorbing C*-algebras
• Asymptotically unitary equivalence and asymptotically inner automorphisms