[Paper Review] Homomorphisms from AH-algebras

Huaxin Lin|arXiv (Cornell University)|Feb 22, 2011

Advanced Operator Algebra Research16 references19 citations

TL;DR

This paper establishes a complete classification criterion for unital monomorphisms from general unital AH-algebras into unital simple C*-algebras with tracial rank at most one. It proves that two such homomorphisms are approximately unitarily equivalent if and only if they agree in KL-theory, induce the same continuous affine map on tracial state spaces, and induce the same homomorphism on K₁ modulo the closure of the image of K₀ in the affine functions on the tracial state space.

ABSTRACT

Let $C$ be a general unital AH-algebra and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. Suppose that $ϕ, ψ: C o A$ are two unital monomorphisms. We show that $ϕ$ and $ψ$ are approximately unitarily equivalent if and only if \beq[ϕ]&=&[ψ] { m in} KL(C,A), ϕ_{\sharp}&=&ψ_{\sharp} and ϕ^†&=&ψ^†, \eneq where $ϕ_{\sharp}$ and $ψ_{\sharp}$ are continuous affine maps from tracial state space $T(A)$ of $A$ to faithful tracial state space $T_{ m f}(C)$ of $C$ induced by $ϕ$ and $ψ,$ respectively, and $ϕ^‡$ and $ψ^‡$ are induced homomorphisms from $K_1(C)$ into $\Aff(T(A))/\bar{ρ_A(K_0(A))},$ where $\Aff(T(A))$ is the space of all real affine continuous functions on $T(A)$ and $\bar{ρ_A(K_0(A))}$ is the closure of the image of $K_0(A)$ in the affine space $\Aff(T(A)).$ In particular, the above holds for $C=C(X),$ the algebra of continuous functions on a compact metric space. An approximate version of this is also obtained. We also show that, given a triple of compatible elements $κ\in KL_e(C,A)^{++},$ an affine map $γ: T(C) o T_{ m f}(C)$ and a \hm $\af: K_1(C) o \Aff(T(A))/\bar{ρ_A(K_0(A))},$ there exists a unital monomorphism $ϕ: C o A$ such that $[h]=κ,$ $h_{\sharp}=γ$ and $ϕ^†=\af.$

Motivation & Objective

• To provide a complete set of invariants that classify unital monomorphisms from general unital AH-algebras into unital simple C*-algebras with tracial rank at most one.
• To extend previous classification results—previously restricted to commutative C(X) or special AH-algebras—to arbitrary unital AH-algebras without requiring slow dimension growth.
• To establish a realization theorem: given compatible data in KL-theory, tracial state maps, and K₁ homomorphisms, there exists a unital monomorphism realizing these invariants.
• To resolve a long-standing technical obstacle in classifying locally AH-algebras with no dimension growth by generalizing Gong’s decomposition theorem to almost multiplicative maps.
• To lay the foundation for proving that unital simple locally AH-algebras with slow dimension growth are classifiable by the Elliott invariant.

Proposed method

• Use the inductive limit structure of AH-algebras $C = \lim_{\to} (B_n, \psi_n)$, where each $B_n$ is a hereditary subalgebra of a matrix algebra over continuous functions on finite CW complexes.
• Construct a sequence of unital monomorphisms $\varphi_n: C_n \to A$ on finite stages $C_n \subset C$, using the compatibility of $\kappa_n = \kappa \circ [\imath_{n,\infty}]$, $\gamma_n = \gamma \circ (\imath_{n,\infty})_\sharp$, and $\alpha_n = \alpha \circ \imath_{n,\infty}^\ddagger$.
• Apply a perturbation result (Lemma 6.10) to produce unitaries $u_n$ such that the conjugated maps $\mathrm{Ad}\,u_n \circ \varphi_n$ are uniformly close on increasing finite sets $\mathcal{F}_n$.
• Define the limit homomorphism $h(f) = \lim_{n\to\infty} \mathrm{Ad}\,u_n \circ \varphi_{n+1} \circ \imath_n(f)$, ensuring $h$ is a unital monomorphism with the desired invariants.
• Use the compatibility of $\kappa$, $\gamma$, and $\alpha$ to ensure consistency across the inductive limit and verify that the resulting $h$ satisfies $[h] = \kappa$, $h_\sharp = \gamma$, and $h^{\ddagger} = \alpha$.
• Leverage the structure of $\mathrm{Aff}(T(A))/\overline{\rho_A(K_0(A))}$ and the continuous affine maps from $T(A)$ to $T_f(C)$ to control the tracial and K-theoretic data.

Experimental results

Research questions

• RQ1 When are two unital monomorphisms $\varphi, \psi: C \to A$ from a unital AH-algebra $C$ to a unital simple $C^*$-algebra $A$ with $TR(A) \leq 1$ approximately unitarily equivalent?
• RQ2 What is the complete set of invariants—beyond $KL$-theory—that classify such homomorphisms?
• RQ3 Can a unital monomorphism realizing given compatible data in $KL(C,A)^{++}$, tracial state maps, and $K_1$-homomorphisms be constructed?
• RQ4 How can Gong’s decomposition theorem be extended to include almost multiplicative maps, enabling classification of locally AH-algebras without dimension growth?
• RQ5 To what extent do tracial state maps and $K_1$-data refine $KL$-theory in the classification of $C^*$-algebra homomorphisms?

Key findings

• Two unital monomorphisms $\varphi, \psi: C \to A$ are approximately unitarily equivalent if and only if $[\varphi] = [\psi]$ in $KL(C,A)$, $\varphi_\sharp = \psi_\sharp$, and $\varphi_\rho = \psi_\rho$, where $\varphi_\sharp$ maps $T(A)$ to $T_f(C)$ and $\varphi_\rho$ maps $K_1(C)$ to $\mathrm{Aff}(T(A))/\overline{\rho_A(K_0(A))}$.
• The result holds for all unital AH-algebras, not just commutative ones or those of slow dimension growth, thus removing a key technical restriction from prior work.
• A realization theorem is proven: given compatible $\kappa \in KL_e(C,A)^{++}$, an affine map $\gamma: T(A) \to T_f(C)$, and a homomorphism $\alpha: K_1(C) \to \mathrm{Aff}(T(A))/\overline{\rho_A(K_0(A))}$, there exists a unital monomorphism $h: C \to A$ with $[h] = \kappa$, $h_\sharp = \gamma$, and $h^{\ddagger} = \alpha$.
• The construction uses an inductive limit approach with unitary conjugations to ensure uniform convergence on dense subsets, yielding a well-defined limit homomorphism.
• The key technical advance is the extension of Gong’s decomposition theorem to almost multiplicative maps, enabling classification of unital simple locally AH-algebras with no dimension growth.
• The results provide a foundational step toward proving that unital simple locally AH-algebras with slow dimension growth are classifiable by the Elliott invariant.