[Paper Review] Some classifiable groupoid C*-algebras with prescribed K-theory

Ian F. Putnam|arXiv (Cornell University)|Nov 15, 2016

Advanced Operator Algebra Research10 references4 citations

TL;DR

This paper constructs minimal, amenable, étale equivalence relations on a Cantor set whose associated groupoid C*-algebras are tracially AF and thus classifiable under the Elliott program. For any given simple, acyclic dimension group $G_0$ and countable, torsion-free abelian group $G_1$, the construction realizes $K_0 \cong G_0$ and $K_1 \cong G_1$, demonstrating that such $K$-theoretic invariants are realizable via étale groupoid C*-algebras.

ABSTRACT

Given a simple, acyclic dimension group $G_{0}$ and countable, torsion-free, abelian group $G_{1}$, we construct a minimal, amenable, étale equivalence relation $R$ on a Cantor set whose associated groupoid $C^{*}$-algebra, $C^{*}(R)$, is tracially AF, and hence classifiable in the Elliott classification scheme for simple, amenable, separable $C^{*}$-algebras, and with $K_{*}(C^{*}(R)) \cong(G_{0}, G_{1})$.

Motivation & Objective

• To determine which classifiable $C^*$-algebras—specifically those satisfying the Elliott classification program—can be realized as groupoid $C^*$-algebras.
• To address the range problem in the Elliott program by constructing $C^*$-algebras with prescribed $K$-theoretic invariants using étale groupoids.
• To show that for any simple, acyclic dimension group $G_0$ and countable, torsion-free abelian group $G_1$, there exists a minimal, amenable, étale equivalence relation on a Cantor set whose $C^*$-algebra has $K_*$-theory isomorphic to $(G_0, G_1)$.
• To establish that the constructed $C^*$-algebra is tracially AF, ensuring its classification within the Elliott framework.

Proposed method

• Constructs a minimal, amenable, étale equivalence relation $R$ on a Cantor set $X$ using inductive limit techniques on finite approximations of the groupoid structure.
• Defines a sequence of finite-dimensional $C^*$-subalgebras $B_l$ within $C^*(R)$, built from projections $\bar{e}_l$ associated with finite paths in the groupoid.
• Uses a decomposition of the groupoid algebra into components indexed by path lengths and uses induction on $l$ to show that $\bar{e}_l$ commutes with finite sets $\mathcal{F}$ up to $\epsilon$-error.
• Applies the tracially AF condition by verifying three key properties: almost-commutation with $\mathcal{F}$, approximation of elements in finite-dimensional subalgebras, and unitary equivalence of $1 - \bar{e}_l$ to a subprojection of a given non-zero projection $p_0$.
• Employs spectral estimates and trace positivity in $C^*$-algebras to ensure that $\bar{e}_l$ becomes unitarily equivalent to a subprojection of $p_0$ for sufficiently large $l$, leveraging the faithfulness of the representation $\pi$.
• Uses the fact that in simple AF-algebras, positivity in $K_0$ is detected by traces, and controls the trace of $\bar{e}_l$ to ensure $\bar{e}_l$ is unitarily equivalent to a subprojection of $p_0$.

Experimental results

Research questions

• RQ1 Can every pair $(G_0, G_1)$, where $G_0$ is a simple, acyclic dimension group and $G_1$ is a countable, torsion-free abelian group, be realized as the $K_*$-theory of a groupoid $C^*$-algebra?
• RQ2 Is it possible to construct a minimal, amenable, étale equivalence relation on a Cantor set whose associated $C^*$-algebra is tracially AF and hence classifiable in the Elliott program?
• RQ3 What conditions on $K$-theory ensure that a $C^*$-algebra arises from an étale groupoid?
• RQ4 How can one ensure that the tracially AF condition is satisfied for a groupoid $C^*$-algebra constructed via inductive limits of finite-dimensional subalgebras?
• RQ5 Can the unitary equivalence of $1 - \bar{e}_l$ to a subprojection of a given projection $p_0$ be guaranteed in such constructions?

Key findings

• For any simple, acyclic dimension group $G_0$ and countable, torsion-free abelian group $G_1$, there exists a minimal, amenable, étale equivalence relation $R$ on a Cantor set such that $C^*(R)$ is tracially AF.
• The $K_0$-group of $C^*(R)$ is isomorphic to $G_0$, and the $K_1$-group is isomorphic to $G_1$, thus realizing the prescribed $K$-theory.
• The construction ensures that $C^*(R)$ satisfies the tracially AF condition, which implies that it is classifiable in the Elliott program.
• The finite-dimensional subalgebras $(1 - \bar{e}_l)B_l(1 - \bar{e}_l)$ approximate elements of $C^*(R)$ up to arbitrary $\epsilon$-error in norm and almost-commute with finite sets.
• The projection $1 - \bar{e}_l$ becomes unitarily equivalent to a subprojection of any given non-zero projection $p_0$ for sufficiently large $l$, due to trace control and spectral estimates.
• The representation $\pi$ of $C^*(R)$ is faithful, and the strong operator limit of $\pi(1 - \bar{e}_l)$ ensures that $\pi((1 - \bar{e}_l)p_0)$ remains non-zero for large $l$, enabling the unitary equivalence argument.