## Notes of Rufus Willett’s second Southampton lecture 30-03-2017

Dynamic asymptotic dimension and applications, II

For towers of finite index subgroups, Yamauchi has shown that ${asdim}$ of box space is either equal to ${asdim(G)}$ or to ${\infty}$.

Let ${G}$ act freely on compact space ${X}$.

1. Back to tower dimension

This is the minimal ${d}$ such that ${X}$ can be ${E}$-covered by ${d+1}$ castles, for all finite ${E\subset F}$. A castle is a union of towers. A tower is a disjoint union ${FU}$, ${U\subset X}$ open, ${F\subset G}$ finite. A castle is a finite collection of disjoint towers. ${E}$-covered means that every point ${x}$ belongs to some ${gU}$ in some tower ${FU}$, and ${Eg^{-1}\subset F}$.

This is inspired from Rokhlin towers. It measures to what extent open pieces on which the action is trivial can be made disjoint. Topological dimension is obviously an obstacle.

2. Nuclear dimension

2.1. Cross-product

The algebraic cross-product of the action is the complex ${\star}$-algebra generated by ${C(X)}$ and operators ${T_g}$, ${g\in G}$ subject to relations ${T_g T_h =T_{gh}}$, ${T_g^*=T_{g^{-1}}}$, ${T_g f T_g^*=f\circ g^{-1}}$.

Complete it for the (maximal) norm

$\displaystyle \begin{array}{rcl} \|a\|=\sup\{\|\pi(a)\|_{B(H)}\,;\,\pi\textrm{ representation in }B(H)\}. \end{array}$

2.2. Nuclear versus tower dimension

Key observation. Let ${(U,F)}$ be a tower and ${\phi}$ a continuous function on ${X}$ supported on ${gU}$, for some ${g\in F}$. Then up to isomorphism, the sub-${C^*}$-algebra generated by elements ${T_h \phi}$, ${h\in F}$, is contained in the easy ${C^*}$-algebra of matrices with entries in ${C_0(U)}$.

Let us call ${c}$-block a ${C^*}$-algebra which is a direct sum of full matrix algebras over ${C_0(X_i)}$ where ${X_i}$‘s are topological spaces of covering dimensions ${\leq c}$.

Say a ${C^*}$-algebra ${A}$ has finite nuclear dimension if ${\exists c,d}$ such that for any finite dimensional ${\star}$-subspace ${F\subset A}$ and ${\epsilon>0}$, there are ${c}$-blocks ${B_0,\ldots,B_d\subset A}$ and contractive completely positive operators ${\pi_i:A\rightarrow B_i}$ such that ${\forall a\in F}$,

$\displaystyle \begin{array}{rcl} \|\sum_i \pi_i(a)-a\|\leq\epsilon\|a\| \end{array}$

We now show that, if the ${G}$ action on ${X}$ has tower-dimension ${d}$ and ${X}$ has covering dimension ${d}$, then the cross-product algebra satisfies the above. We use a “flat” partition of unity ${(\phi_i)}$, i.e. with small commutators ${[\phi_i,T_h]}$ for ${h\in E}$. This shows

Theorem 1 If the ${G}$ action on ${X}$ has finite tower-dimension and ${X}$ has finite covering dimension, then the cross-product algebra has finite nuclear dimension.

3. K-theory

If the ${c}$-blocks above were ideals, we could compute the K-theory of the cross-product from the K-theory of blocks, by Mayer-Vietoris. But they are not.

Oyono-Oyono and Yu developped a version of K-theory which gives one Mayer-Vietoris-like sequences for subalgebras with the following ideal-like property: for every finite subset ${E\subset G}$, ${C^*(EBE)}$ is still a ${c}$-block. This allows to prove Baum-Connes conjecture and Kunneth formula the cross-product (Guentner-Willett-Yu).