Notes about math, research, and more.
:tags: research

Coarse geometry

Paper: Quasi-Locality for etale Groupoids

  • Authors: Jian, Zhang, and Zhang

Notes from EGd

Big idea would be can we relax the amenability requirement on some of their results. For example Theorem 5.1:

Theorem 5.1: Let \(\mathcal{G}\) be a locally compact, \(\sigma\text{-compact}\) and etale groupoid. If \(\mathcal{G}\) is amenable, then we have \(C_{r}^*(\mathcal{G}) = C_{u}^*(\mathcal{G})^\mathcal{G} = C_{uq}^*(\mathcal{G})^\mathcal{G}\).

Problem:

  • Amenability is so nice that removing it can be messy quickly
  • Find some groupoid-theoretic version of uniform CEH and see what happens to the \(C^\ast\text{-algebra}\).
    • Lemma 5.5 but uniform CEH instead of property A to prove a more general Theorem 5.1
    • Finding groupoids

Paper: A Gelfand-type duality for coarse metric spaces with property A

  • Authors: Braga and Vignati

Notes from EGd

  • What could we do if spaces don’t have property A?
  • Can we find counterexamples (are my graphs counter-examples)

Review:

  • Main purpose

    Prove two results for a given uniformly locally finite metric space with property A:

    1. The group of outer automorphisms of its uniform Roe algebra is isomorphic to its group of bijective coarse equivalences module closeness.
    2. The group of outer automorphisms of its Roe algebra is isomorphic to its group of coarse equivalences modulo closeness.

Q for EG Gelfand duality:

  • Proved every comm \(C^\ast\text{-algebra}\) is C0(X).
  • Gelfand transform is a functor; a homomorphism between 2 \(C^\ast\text{-algebras}\)
  • non-uniform Roe algebra: elements in the matrix are compact operators on a Hilbert space instead of just complex numbers.
  • Same k-theory

Isomorphism of uniform Roe algebras implies bijective coarse equivalence

From email EG-> PK: To be honest I think we might do better to look a little farther afield than Property A; for example, the attached paper by Sako (already almost 5 years old) has some pretty thorough characterizations of the C*-algebraic properties of the Roe algebra when the space has property A. Namely, C*u(X) is nuclear iff it’s exact iff X has A.

With that in mind, one possible project is the following: For the spaces you’ve constructed which CEH but don’t have (A), could we prove that an isomorphism of uniform Roe algebras implies a bijective coarse equivalence? This is related to the BBFVW paper (they prove that this holds for expanders), and it’s also related to Corollary 3.5 (cf also Remark 3.7) in the BBFKVW paper.

Question based on Li-Liao-Winter’s paper “The diagonal dimension of sub \(C^\ast\text{-algebras}\)”

Background

There’s a refined version of nuclearity called “nuclear dimension” and a refinement is “diagonal dimension” - introduced in the above paper.

LLW relate diagonal dimension for Roealgebras with the asymptoticdimension of the underlying space \(X\). They also relate diagonal dimension for groupoid \(C^\ast\text{-algebras}\) with something called “dynamical aysmptotic dimension (dad)” of the groupoid.

groupoid groupoid

groupoid groupoid

groupoid

groupoid

Possible questions

  1. Remark 6.8: they say that they would expect dad(G) to equal the diagonal dimension if the groupoid is principal and ample.
  2. Question 6.11: The nuclear dimension of a groupoid \(C^\ast\text{-algebra}\) in large generality is bounded by a term of the form \(dad^{+1}(\mathcal{G}) \cdot dim^{+1}(\mathcal{G}^{(0)}\), and it would be interesting to see whether along the same lines one can also obtain an upper bound for diagonal dimension. More precisely: Is it true that for any etale principal groupoid \(\mathcal{G}\) we have the estimate

\[ dim^{+1}(\mathcal{G})(C_{0}(\mathcal{G})) \subset C_{r}^\ast(\mathcal{G}) \leq dad^{+1}(\mathcal{G}^{(0)} ? \]

Graph \(C^\ast\text{-algebras}\)

Project: Can we describe the exotic Cuntz algebra as a k-graph algebra?

  • Jeff Boersema, Efren Ruiz, and Peter Stacey proved that for odd n, there are two real C*-algebras [coefficient field R instead of C] whose complexification is the Cuntz algebra On. Jeff later proved that the exotic Cuntz algebra [the unexpected one] cannot be a graph C*-algebra.
  • With Jeff Boersema and Sarah Browne, I proved that the exotic Cuntz algebra is stably isomorphic to a [real] 3-graph algebra. The 3-graph we constructed has infinitely many vertices, and exotic On is unital, so they can’t be on-the-nose isomorphic.
  • So, can we find a k-graph with finitely many vertices which will give us exotic On?

Project: Can we prove a k-graph version of Cuntz splice?

  • Eilers, Restorff, Ruiz, and Sorensen proved that for directed graphs, gluing on a certain graph [Cuntz splice] doesn’t change the C*-algebra
  • The proof that doing Cuntz splice twice doesn’t change the C*-algebra is graph-theoretic and constructive; the proof that doing Cuntz splice once is the same as doing Cuntz splice twice relies on some K-theory magic.
  • With past summer research teams, we’ve made some progress towards extending the Cuntz-splice-twice part of the proof to k-graphs, but more along the lines of clarifying what exactly is the question than completing the proof. And I have not tyought about a k-graph version of the K-theory-magic part of the proof

Machine learning

  • Really interesting, but I don’t think a nice fit for us.