## Notes of Kang Li’s Southampton lecture 29-03-2017

Structure and ${K}$-theory of uniform Roe algebras

It turns out that uniform Roe algebras are easier to classify than Roe algebras.

1. Uniform Roe algebra of a space

See Wright’s talk for the definition: this is the closure (in operator norm on ${\ell^2}$) of finite propagation operators, on a bounded geometry locally finite metric space. It contains ${\ell^\infty}$ as diagonal matrices.

Theorem 1 (Brodski-Niblo-Wright 2007) For coarsely equivalent spaces, URAs get isomorphic once tensored with ${K}$. The converge holds for spaces with property (A) (Spakula-Willett 2013).

Need to tensor with ${K}$ arises because coarse equivalences are not bijections. Bijective coarse equivalences lead to isomorphisms of URAs.

Proposition 2 (Willett) For bijectively coarsely equivalent spaces, URAs with their ${\ell^\infty(X)}$ lead to equivalent Cartan pairs.

Question. If groups have isomorphic URAs, can one conclude that they are bijectively coarsely equivalent ?

Answer is positive for non-amenable exact, or property (A) countable groups, by Spakula-Willett’s result combined with Whyte’s upgrading procedure.

Answer is positive for finite groups. Indeed, for finite metric spaces ${X}$, ${URA(X)}$ is the full matrix algebra ${M_{|X|}({\mathbb C})}$.

With my office-mate Liao, we treat the locally finite case (every finitely generated subgroup is finite). Indeed, in this case, the URA is locally AF (Willett). Locally AF algebras are classified by their ${K_0}$ groups (Elliott’s theorem).

Theorem 3 (Li-Liao) Two countable locally finite groups are bijectively coarsely equivalent iff they have isomorphic URAs.

2. Supernatural numbers

Definition 4 For ${p}$ prime, let ${n_p}$ be the largest integer ${m}$ such that ${p^m}$ divides the order of some finite subgroup of ${G}$. Define the supernatural numberof ${G}$ by ${s(G)=\prod_{p} p^{n_p}}$ (formal product).

For instance, ${s(\bigoplus {\mathbb Z}/2{\mathbb Z})=2^\infty}$, ${s({\mathbb Q}/{\mathbb Z})=2^\infty 3^\infty\cdots}$.

Theorem 5 (Protasov 2002) Two countable locally finite groups are bijectively coarsely equivalent iff they have the same supernatural number.

Theorem 6 For a countable group ${G}$, the following are equivalent.

1. ${URA(G)}$ is AF (i.e. an inductive limit of finite dimensional ${C^*}$ algebras).
2. ${URA(G)}$ is locally AF (i.e. every finite subset is arbitrarily close to some finite dimensional sub ${C^*}$ algebra).
3. ${URA(G)}$ is finite (${v^*v=1 \Rightarrow vv^*=1}$).
4. ${G}$ is locally finite.

The last equivalence is due to Scarparo (2016).

Note that Smith showed in 2006 that ${G}$ is locally finite iff ${asdim(G)=0}$, so we investigate a metric version of previous theorem.

Theorem 7 (Li-Willett 2017) Let ${X}$ be a locally finite bounded geometry metric space. Then the following are equivalent.

1. ${URA(X)}$ is AF.
2. ${URA(X)}$ is locally AF.
3. ${asdim(X)=0}$.

These properties imply that ${URA(X)}$ is finite, but the converse need not hold.

2. Nuclear dimension

In 2010, Winter and Zacharias showed that for arbitrary ${C^*}$ algebras,${dim_{nuc}A=0}$ iff ${A}$ is locally AF. More generally, ${dim_{nuc}URA(X)\leq asdim(X)}$. Does equality hold ? Answer is positive for dimensions 0 and 1.

3. Asymptotic dimension 1

For non-amenable metric spaces of ${asdim=1}$, ${K_0(URA(X))=0}$, hence ${K_0}$ does not help. There exist non coarsely equivalent groups with ${asdim=1}$ (eg ${F_2}$ and ${F\wr F_2}$ with ${F}$ finite.

In 2016, Elliott and Sierakowski asked wether this would hold in general. Answer is negative. With Willett, we show that for a genus 2 surface group, ${K_0(URA(G))\not=0}$.