Tag Archives: Roe algebra

Maximal Roe algebras, part 2

Let \(X\) be a bounded geometry metric space.  At the end of the previous post, we observed that if \( \pi \colon {\mathbb C}[X] \to {\mathfrak B}(H) \) is a Hilbert space representation of the translation algebra of \(X\), then any unit vector in the range of one of the projections \(\pi(V_{x,x})\) corresponding to a point of \(X\) generates a subrepresentation isomorphic to the regular one.  It follows that if \(\pi\) does not contain a copy of the regular representation, then the projections \(\pi(V_{x,x})\) must be zero for every \(x\in X\).

Surprisingly enough, such representations do exist! Continue reading

Thinking about “maximal Roe algebras”

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