A correction to EOTAM

In my book Elliptic operators, topology and asymptotic methods (both the first and the second editions) I give a discussion of the representation theory of the groups Spin and Pin which was based (as far as I can now remember) on some notes that I took when I attended Adams’ famous course on the exceptional Lie groups, as a Part III student in 1981.  I no longer seem to have those, unfortunately (although meanwhile a version of Adams’ own notes on his approach has been published by University of Chicago Press).  Meanwhile, in 2010 Darij Grinberg pointed out on Math Overflow that the argument I gave was garbled: see this link.  In this post I want to explain what is garbled and how the useful part of the argument can be recovered.  Continue reading

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