[Paper Review] Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras

Huaxin Lin|ArXiv.org|Jun 3, 2008

Advanced Operator Algebra Research31 references9 citations

TL;DR

This paper establishes a complete classification of unital separable simple amenable C*-algebras with tracial rank at most one by proving that two unital monomorphisms between such algebras are asymptotically unitarily equivalent if and only if they agree on KK-theory, trace maps, and a rotation-related map. The result extends the Elliott classification program to a broader class of C*-algebras, including Z-stable AH-algebras isomorphic to those with no dimension growth.

ABSTRACT

Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $ϕ_1, ϕ_2: C o A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t o\infty}u_t^*ϕ_1(c)u_t=ϕ_2(c) foral c\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $ϕ_1^‡=ϕ_2^‡,$ $(ϕ_1)_T=(ϕ_2)_T$ and a rotation related map $\bar{R}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\cal A}$ of unital separable simple amenable \CA s which is strictly larger than the class of separable \CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One consequence of the main result is that all unital simple AH-algebras which are ${\cal Z}$-stable are isomorphic to ones with no dimension growth.

Motivation & Objective

• To establish a uniqueness theorem for asymptotic unitary equivalence of unital monomorphisms between simple amenable C*-algebras with tracial rank ≤1.
• To extend the Elliott classification program to a strictly larger class of C*-algebras beyond those with tracial rank 0 or 1.
• To prove that Z-stable unital simple AH-algebras are isomorphic to ones with no dimension growth, using asymptotic unitary equivalence.
• To characterize a class A of C*-algebras closed under tensoring with UHF algebras, inductive limits, and tensor products, including many ASH-algebras and Z-stable algebras.
• To provide a framework for classifying C*-algebras via KK-theory, traces, and a new rotation map, enabling classification beyond tracial rank zero.

Proposed method

• Use of the Basic Homotopy Lemma for C*-algebras with tracial rank one, recently established in [38], to control perturbations of homomorphisms.
• Application of the Universal Coefficient Theorem (UCT) to ensure that KK-theory classes lift to actual homomorphisms.
• Construction of a continuous path of unitaries in A that asymptotically intertwine two unital monomorphisms φ₁ and φ₂.
• Verification of asymptotic unitary equivalence via four invariants: [φ₁] = [φ₂] in KK(C,A), φ₁‡ = φ₂‡, (φ₁)T = (φ₂)T, and R̄φ₁,φ₂ = 0.
• Leveraging W. Winter’s method involving tensoring with UHF algebras Mₚ to reduce classification to algebras with tracial rank zero.
• Use of inductive limit and tensor product closure properties to define and characterize the class A of C*-algebras.

Experimental results

Research questions

• RQ1 When are two unital monomorphisms φ₁, φ₂: C → A asymptotically unitarily equivalent for simple amenable C*-algebras with tracial rank ≤1?
• RQ2 What K-theoretic and tracial invariants are necessary and sufficient to classify such homomorphisms up to asymptotic unitary equivalence?
• RQ3 Can Z-stable unital simple AH-algebras be classified as isomorphic to those with no dimension growth?
• RQ4 What is the minimal set of invariants (beyond KK-theory and traces) required to classify C*-algebras in the Elliott program beyond tracial rank zero?
• RQ5 Which C*-algebras belong to the class A closed under tensor products, inductive limits, and tensoring with UHF algebras, and what is their structural characterization?

Key findings

• Two unital monomorphisms φ₁, φ₂: C → A are asymptotically unitarily equivalent if and only if [φ₁] = [φ₂] in KK(C,A), φ₁‡ = φ₂‡, (φ₁)T = (φ₂)T, and the rotation map R̄φ₁,φ₂ = 0.
• The class A of unital separable simple amenable C*-algebras includes all unital simple AH-algebras, all Z-stable algebras with TR ≤1, and is closed under tensor products, inductive limits, and tensoring with UHF algebras.
• All unital simple AH-algebras that are Z-stable are isomorphic to ones with no dimension growth, a key consequence of the main classification result.
• Tensor products of algebras in A remain in A, and A is closed under inductive limits, confirming its robustness for classification purposes.
• The class A includes unital simple ASH-algebras with arbitrary metrizable Choquet simple tracial state spaces and K₀-group ℤ, as well as those with non-Riesz K₀-groups.
• For A, B ∈ A, A ⊗ ℤ ≅ B ⊗ ℤ if and only if Ell(A ⊗ ℤ) ≅ Ell(B ⊗ ℤ), providing a complete invariant for Z-stable algebras in A.