Tag Archives: incompetence

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.