# Traces and commutators

The following is a true (and well-known) theorem: $$\newcommand{\Tr}{\mathop{\rm Tr}}$$

Suppose $$A$$ and $$B$$ are bounded operators on a Hilbert space, and $$AB$$ and $$BA$$ are trace class.  Then $$\Tr(AB)=\Tr(BA)$$.

This is easy to prove if one of the operators $$A,B$$ is itself of trace class, or if they both are Hilbert-Schmidt (the obvious calculation works).  In the general case it is a bit harder.  The “usual” argument proceeds via Lidskii’s trace theorem – the trace of any trace-class operator is the sum of the eigenvalues – together with the purely algebraic fact that the nonzero eigenvalues of $$AB$$ and $$BA$$ are the same (including multiplicities).  Continue reading

# 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

# 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

# The Peter Principle Revisited

Not really “coarse mathematics” in the sense I initially intended, but feels rather relevant to the work of a department head.   This paper uses a simulation to demonstrate that, under certain hypotheses, a “hierarchical” organization will function more efficiently by promoting people at random than by always promoting those who are most competent in their current position.