[Paper Review] Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras

Huaxin Lin, Zhuang Niu|ArXiv.org|Feb 11, 2008

Advanced Operator Algebra Research13 references4 citations

TL;DR

This paper establishes a complete classification of unital, simple, Z-stable C*-algebras with tracial rank zero by solving the lifting problem for KK-elements and characterizing asymptotic unitary equivalence of unital monomorphisms. It proves that unital monomorphisms from a unital AH-algebra $ C $ to a unital simple C*-algebra $ A $ with tracial rank zero are classified up to asymptotic unitary equivalence by their KK-class, induced tracial state map, and rotation map, leading to a complete isomorphism classification for Z-stable C*-algebras with isomorphic Elliott invariants.

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 $κ$ for which $κ(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}$ and $κ([1_C])=[1_A]$. Suppose that $κ\in {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $ϕ: C o A$ such that $[ϕ]=κ.$ Suppose that $C$ is a unital AH-algebra and $λ: \mathrm{T}(A) o \mathrm{T}_{\mathtt{f}}(C)$ is a continuous affine map for which $τ(κ([p]))=λ(τ)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $τ\in \mathrm{T}(A),$ 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 $ϕ: C o A$ such that $ϕ$ induces both $κ$ and $λ.$ Suppose that $h: C o A$ is a unital monomorphism and $γ\in \mathrm{Hom}(\Kone(C), \aff(A)).$ We show that there exists a unital monomorphism $ϕ: C o A$ such that $[ϕ]=[h]$ in ${KK}(C,A),$ $τ\circ ϕ=τ\circ h$ for all tracial states $τ$ and the associated rotation map can be given by $γ.$ Applications to classification of simple C*-algebras are also given.

Motivation & Objective

• To resolve the open problem of lifting KK-elements to unital monomorphisms in the context of C*-algebras with tracial rank zero.
• To characterize asymptotic unitary equivalence of unital monomorphisms from a unital AH-algebra $ C $ to a unital simple C*-algebra $ A $ with tracial rank zero.
• To remove restrictions on $ K $-theory and projections separating traces in the classification of simple C*-algebras.
• To establish a complete classification theorem for unital, separable, simple, $ \mathcal{Z} $-stable C*-algebras using the Elliott invariant and tracial rank conditions.

Proposed method

• Prove that for unital simple C*-algebras $ A $ and $ C $ with tracial rank zero and $ C $ amenable satisfying the UCT, any $ \kappa \in {KK}_e(C,A)^{++} $ lifts to a unital monomorphism $ \phi: C \to A $ with $ [\phi] = \kappa $.
• Establish a bijection between asymptotic unitary equivalence classes of unital monomorphisms $ C \to A $ and the set $ (KKT(C,A)^{++}, \mathrm{Hom}(K_1(C), \mathrm{Aff}(T(A)))/\mathcal{R}_0) $, where $ \mathcal{R}_0 $ is a subgroup of vanishing rotation maps.
• Use the theory of asymptotic unitary equivalence from [15] to relate the classification to the full data of $ KK $-class, tracial state map, and rotation map.
• Apply the lifting result to show that two unital $ \mathcal{Z} $-stable C*-algebras are isomorphic if their Elliott invariants are isomorphic and their tensor products with any UHF-algebra have tracial rank zero.
• Leverage results from Winter [24] and the structure of inductive limits of type I C*-algebras with unique tracial states to extend classification beyond finitely generated $ K $-theory.

Experimental results

Research questions

• RQ1 Can every $ \kappa \in {KK}_e(C,A)^{++} $ be realized as the $ KK $-class of a unital monomorphism $ \phi: C \to A $, when $ A $ and $ C $ are unital simple C*-algebras with tracial rank zero and $ C $ satisfies the UCT?
• RQ2 Under what conditions are two unital monomorphisms $ \phi, \psi: C \to A $ asymptotically unitarily equivalent?
• RQ3 Can a given rotation map $ \gamma \in \mathrm{Hom}(K_1(C), \mathrm{Aff}(T(A))) $ be realized by a unital monomorphism $ \phi: C \to A $ with fixed $ KK $-class and tracial state map?
• RQ4 Does the isomorphism of Elliott invariants imply isomorphism for unital, separable, simple $ \mathcal{Z} $-stable C*-algebras that are inductive limits of type I algebras with unique tracial states?
• RQ5 Is the classification of unital $ \mathcal{Z} $-stable C*-algebras complete when tensoring with any UHF-algebra yields tracial rank zero?

Key findings

• Every $ \kappa \in {KK}_e(C,A)^{++} $ lifts to a unital monomorphism $ \phi: C \to A $, solving a long-standing lifting problem for C*-algebras with tracial rank zero.
• Asymptotic unitary equivalence classes of unital monomorphisms $ C \to A $ are in bijection with the set $ (KKT(C,A)^{++}, \mathrm{Hom}(K_1(C), \mathrm{Aff}(T(A)))/\mathcal{R}_0) $, providing a complete classification invariant.
• Two unital $ \mathcal{Z} $-stable C*-algebras are isomorphic if their Elliott invariants are isomorphic and their tensor products with any UHF-algebra have tracial rank zero.
• For unital, separable, simple $ \mathcal{Z} $-stable C*-algebras that are inductive limits of type I algebras with unique tracial states, isomorphism is equivalent to having isomorphic Elliott invariants.
• The classification extends beyond finitely generated $ K $-theory and projections separating traces, removing key restrictions from prior results in the literature.