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.