Tag Archives: Roe algebra

Maximal Roe algebras, part 2

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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”

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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 →