[Paper Review] The classification of simple separable unital locally ASH algebras

George A. Elliott, Guihua Gong+2 more|arXiv (Cornell University)|Jun 7, 2015

Advanced Operator Algebra Research27 references3 citations

TL;DR

This paper establishes that simple separable unital locally approximately subhomogeneous (locally ASH) C*-algebras are classifiable via the Elliott invariant when Jiang-Su stable. By showing that $A \otimes Q$ is tracially approximated by unital Elliott-Thomsen algebras with trivial $\mathrm{K}_1$-group—using noncommutative cell complexes and the Winter deformation technique—it follows that such algebras are classified by their naive Elliott invariant, completing a key step in the classification program for C*-algebras.

ABSTRACT

Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $A\otimes Q$ can be tracially approximated by unital Elliott-Thomsen algebras with trivial $ extrm{K}_1$-group, where $Q$ is the universal UHF algebra. In particular, it follows that $A$ is classifiable by the Elliott invariant if $A$ is Jiang-Su stable.

Motivation & Objective

• To establish the classification of simple separable unital locally ASH C*-algebras using the Elliott invariant.
• To show that $A \otimes Q$ is tracially approximated by unital Elliott-Thomsen algebras with trivial $\mathrm{K}_1$-group for any locally ASH algebra $A$.
• To prove that Jiang-Su stable locally ASH algebras are classifiable by the naive Elliott invariant, leveraging recent advances in nuclear dimension and noncommutative cell complexes.
• To extend the classification framework to include C*-algebras arising from minimal homeomorphisms with mean dimension zero.

Proposed method

• Utilizes noncommutative cell complexes (NCCCs) as building blocks to locally approximate unital subhomogeneous C*-algebras.
• Applies the tracial approximation framework (TA) to show $A \otimes Q \in \mathrm{TA}\mathcal{C}_0$, where $\mathcal{C}_0$ is the class of unital Elliott-Thomsen algebras with trivial $\mathrm{K}_1$-group.
• Employs the Winter deformation technique, refined by Lin and Niu, to establish isomorphism in the presence of UCT and finite nuclear dimension.
• Relies on the result from [5] that Jiang-Su stable simple unital ASH algebras have finite nuclear dimension, enabling their inclusion in the class studied by Gong, Lin, and Niu.
• Uses pullback diagrams involving $\mathrm{M}_k(\mathrm{C}(S^{n-1}))$ and $\mathrm{M}_k(\mathrm{C}(D^n))$ to construct NCCCs recursively.
• Applies the trace approximation condition: $\sup\{ |\tau(\varrho_i \sigma_i(a) - a)| \} \to 0$ for all $a \in A$, $\tau \in \mathrm{T}(A)$, as $i \to \infty$.

Experimental results

Research questions

• RQ1 Can locally ASH C*-algebras be classified by the Elliott invariant, particularly when Jiang-Su stable?
• RQ2 Is the tensor product $A \otimes Q$ tracially approximated by unital Elliott-Thomsen algebras with trivial $\mathrm{K}_1$-group for any simple separable unital locally ASH algebra $A$?
• RQ3 Does the finite nuclear dimension of Jiang-Su stable ASH algebras imply their embeddability into the class of algebras classifiable via the Winter deformation technique?
• RQ4 Can the classification result be extended to C*-algebras from minimal homeomorphisms with mean dimension zero?

Key findings

• For any simple separable unital locally ASH C*-algebra $A$, the tensor product $A \otimes Q$ belongs to the class $\mathrm{TA}\mathcal{C}_0$, where $\mathcal{C}_0$ consists of unital Elliott-Thomsen algebras with trivial $\mathrm{K}_1$-group.
• It follows that if $A$ is Jiang-Su stable, then $A$ is classifiable by the naive Elliott invariant, including the ordered $\mathrm{K}_0$-group, the class of the unit, the tracial state space, and $\mathrm{K}_1$.
• The nuclear dimension of $A \otimes Q$ is at most 2, which implies finite nuclear dimension and enables the application of the tracial approximation machinery.
• The result recovers the classification of crossed product C*-algebras $\mathrm{C}(X) \rtimes_\sigma \mathbb{Z}$ when $\sigma$ is a minimal homeomorphism of mean dimension zero.
• For any locally ASH algebra $A$, the algebra $A \otimes \mathcal{Z}$ is itself an ASH algebra, and if $A$ is locally AH, then $A \otimes \mathcal{Z}$ is AH.
• The converse of the classification result holds: if $A$ is classifiable by the Elliott invariant and Jiang-Su stable, then $A$ is locally ASH.