One of the things that has happened in coarse geometry while I was busy being department chair is a bunch of papers about “maximal Roe algebras” (some references at the end). Of course these are objects that I feel I ought to understand, so I spent some time trying to figure out the basics.

Let \(X\) be a bounded geometry uniformly discrete metric space. (Something like bounded geometry seems to be necessary, for a reason that I’ll explain below.) We know how to form the *translation algebra* \( {\mathbb C}[X] \) (the *-algebra of finite-propagation matrices on \(X\) ), and this has an obvious representation (the *regular representation*) on \(\ell^2(X)\). Then the usual version of the (uniform) *Roe algebra* is just the C*-algebra obtained by completing \({\mathbb C}[X]\) in this representation. Because it involves only the regular representation we may call this the *reduced* Roe algebra (in analogy to the group case).

Again proceeding by analogy, we should also define the *maximal* Roe algebra by completing \({\mathbb C}[X]\) in the “universal” representation: that is, the *maximal norm* for \(T\in {\mathbb C}[X]\) is

\[ \|T\|_{\max} = \sup \{ \|\pi(T)\| : \pi \in {\mathscr R} \} , \]

where \(\mathscr R\) is the set of all *-representations of \({\mathbb C}[X]\) on a specified (infinite-dimensional) Hilbert space. For this definition to make sense one needs to ensure that the supremum is finite: this follows from a standard “partial translation decomposition” argument (needing bounded geometry), which shows that every element of \({\mathbb C}[X]\) is a *finite* \(\ell^\infty(X)\)-linear combination of partial isometries, together with the fact that the norm of a partial isometry is at most 1 in any representation.

At first it is difficult to think of examples of non-regular representations. In fact, consider the following argument: let \(\pi\) be any Hilbert space representation of \({\mathbb C}[X]\), and consider the partial isometries \(V_{xy}\) in \({\mathbb C}[X]\) which have 1 at the \( (x,y)\) position and 0 elsewhere. (Our space is coarsely connected so these are in the translation algebra.) The \(V_{x,x}\) are projections; let \(\xi_x\) be a unit vector in the range of \(\pi (V_{x,x})\). For all other \(y\in X\) let \(\xi_y = \pi (V_{y,x})\xi_x\). Then the properties of the \(V_{x,y}\) (basically, that they are a system of matrix units) easily show that the \(\xi\)’s span a subrepresentation isomorphic to the regular representation. If it’s not the whole thing, split it off and continue by induction. Hence every representation is just a direct sum of copies of the regular one (??).

I’ll post part 2 after lunch, so, if anyone is reading this in real time, you have a couple of hours to figure out what is wrong with this argument!

#### References

Gong, Guihua, Qin Wang, and Guoliang Yu. 2008. “Geometrization of the Strong Novikov Conjecture for residually finite groups.” *Journal für die reine und angewandte Mathematik (Crelles Journal)* 2008(621):159–189. Retrieved August 13, 2013.

Špakula, Ján, and Rufus Willett. 2013. “Maximal and reduced Roe algebras of coarsely embeddable spaces.” *Journal für die reine und angewandte Mathematik (Crelles Journal)* 2013(678):35–68. Retrieved August 14, 2013.

Willett, Rufus, and Guoliang Yu. 2012a. “Higher index theory for certain expanders and Gromov monster groups, I.” *Advances in Mathematics* 229(3):1380–1416. Retrieved March 14, 2013.

Willett, Rufus, and Guoliang Yu. 2012b. “Higher index theory for certain expanders and Gromov monster groups, II.” *Advances in Mathematics* 229(3):1762–1803. Retrieved March 14, 2013.