Tag Archives: Roe algebra

Positive scalar curvature partial vanishing theorems and coarse indices

This paper, http://arxiv.org/abs/1210.6100, has been accepted by the Proceedings of the Edinburgh Mathematical Society.  I just sent off the copyright transfer form this evening, so everything is now set, I hope.

The paper is mostly paying an expository debt.  In my CBMS lecture notes I said that if one has the Dirac operator on a complete spin manifold \(M\), and if there is some subset \(N\subseteq M\) such that \(D\) has uniformly positive scalar curvature outside \(N\), then the index of \(D\) belongs to the K-theory of the ideal \(I_N \triangleleft C^*(M) \) associated to the subset \(N\).  A very special case of this is the observation of Gromov-Lawson that \(D\) is Fredholm if we have uniformly positive scalar curvature outside a compact set.  There are of course analogous results using thepositivity of the Weitzenbock curvature  term for other generalized Dirac operators.

Until now, I had not written up the proof of this assertion, but I felt last year that it was (past) time to do so.  This paper contains the proof and also that of the associated general form of the Gromov-Lawson relative index theorem which also appears in my CBMS notes. The latter proof uses some results from my paper with Paul Siegel on sheaf theory and Pashcke duality.

The submission to PEMS is in honor of a very pleasant sabbatical spent in Edinburgh in fall 2004.

 

Maximal Roe algebras, part 3

It is a well-known fact that if a group \(\Gamma\) is amenable then the canonical map \(C^*_{\max}(\Gamma) \to C^*_r(\Gamma) \) is an isomorphism.  In this post I’ll follow Spakula and Willett in proving the coarse analog of this statement: for a property A space the canonical map from the maximal to the reduced Roe algebra is an isomorphism.  (The converse is unknown, unlike in the group case.) Continue reading

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