[Paper Review] Classifying $^*$-homomorphisms I: Unital simple nuclear $C^*$-algebras

José R. Carrión, James Gabe+3 more|arXiv (Cornell University)|Jul 12, 2023

Advanced Operator Algebra Research4 citations

TL;DR

This paper provides a new, self-contained classification of unital embeddings from unital separable nuclear C*-algebras satisfying the UCT into unital simple nuclear C*-algebras that absorb the Jiang–Su algebra. Using a refined invariant combining K-theory and tracial data, it establishes a bijective correspondence between approximate unitary equivalence classes of embeddings and morphisms of this total invariant, offering a streamlined proof of the stably finite case of the unital classification theorem in C*-algebra theory.

ABSTRACT

We classify the unital embeddings of a unital separable nuclear $C^*$-algebra satisfying the universal coefficient theorem into a unital simple separable nuclear $C^*$-algebra that tensorially absorbs the Jiang--Su algebra. This gives a new and essentially self-contained proof of the stably finite case of the unital classification theorem: unital simple separable nuclear $C^*$-algebras that absorb the Jiang--Su algebra tensorially and satisfy the universal coefficient theorem are classified by Elliott's invariant of $K$-theory and traces.

Motivation & Objective

• To provide a new, conceptually simpler and self-contained proof of the stably finite case of the unital classification theorem for C*-algebras.
• To classify unital *-homomorphisms from a unital separable nuclear C*-algebra satisfying the UCT into a unital simple nuclear C*-algebra that is Z-stable.
• To introduce and utilize a total invariant $\underline{K}T_u$ combining enriched K-theoretic and tracial data to parametrize unital embeddings up to approximate unitary equivalence.
• To establish that the tracial component of the invariant detects the dichotomy between purely infinite and stably finite C*-algebras, with traces existing precisely in the stably finite case.
• To unify the stably finite and purely infinite cases by proving the stably finite counterpart, completing the full unital classification theorem when combined with the Kirchberg–Phillips theorem.

Proposed method

• Define the total invariant $\underline{K}T_u$ as a combination of $K$-theory and traces, enriched with Bockstein operations and compatibility conditions.
• Use the universal multicoefficient theorem (UMCT) for $KK$-theory to relate morphisms in $KK$-theory to homomorphisms of the invariant $\underline{K}T_u$.
• Leverage the $\sigma$-unital and separable approximation techniques to extend results from separable to non-separable $C^*$-algebras via inductive limits.
• Apply the Kasparov product and natural identifications $KK(\mathbb{C}, S^i I) \cong K_i(I)$ to construct the map $\Gamma^{(A,I)}$ from $KK(A,I)$ to $\mathrm{Hom}(K_*(A), K_*(I))$.
• Use the $\Lambda$-module structure on $\underline{K}(A)$ and $\underline{K}(I)$ to define $\mathrm{Hom}_\Lambda(\underline{K}(A), \underline{K}(I))$, enabling compatibility with Bockstein operations.
• Prove that when $A$ satisfies the UCT, the map $\tilde{\Gamma}_\Lambda^{(A,I)}: KL(A,I) \to \mathrm{Hom}_\Lambda(\underline{K}(A), \underline{K}(I))$ is an isomorphism, enabling classification via the invariant.

Experimental results

Research questions

• RQ1 How can the classification of unital embeddings into Z-stable C*-algebras be achieved using a unified, self-contained framework?
• RQ2 What is the precise role of the total invariant $\underline{K}T_u$ in parametrizing unital *-homomorphisms up to approximate unitary equivalence?
• RQ3 How does the tracial data in $\underline{K}T_u$ detect the stably finite versus purely infinite dichotomy in simple nuclear C*-algebras?
• RQ4 In what way does the universal multicoefficient theorem extend to non-separable $C^*$-algebras, and how does it support the classification result?
• RQ5 Can the stably finite case of the unital classification theorem be proven independently of finite nuclear dimension, using only $\mathcal{Z}$-stability and the UCT?

Key findings

• The classification of unital embeddings $A \hookrightarrow B$ is bijective with morphisms of the total invariant $\underline{K}T_u$ that map faithful traces on $A$ to traces on $B$.
• The invariant $\underline{K}T_u$ is constructed from enriched $K$-theory and traces, incorporating Bockstein operations and compatible $\Lambda$-module structures.
• The map $\Gamma_{\Lambda}^{(A,I)}: KK(A,I) \to \mathrm{Hom}_\Lambda(\underline{K}(A), \underline{K}(I))$ is an isomorphism when $A$ satisfies the UCT, enabling classification via $KL$-theory.
• The proof establishes that $Z_{KK(A,I)}$ maps to zero under $\Gamma_{\Lambda}^{(A,I)}$, ensuring well-definedness of the invariant map.
• The result provides a new, shorter, and more conceptual proof of the stably finite case of the unital classification theorem.
• The classification is achieved by combining the new stably finite result with the Kirchberg–Phillips theorem for the purely infinite case, yielding the full unital classification theorem.