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). Continue reading