  Our motivating question was whether all traces on a U-ultrapower of a
C*-algebra A, where U is a non-principal ultrafilter on N, are necessarily
U-limits of traces on A. We show that this is false so long as A has infinitely
many extremal traces, and even exhibit a 2^c size family of such traces on the
ultrapower. For this to fail even when A has finitely many traces implies that
A contains operators that can be expressed as sums of n+1 but not n
*-commutators, for arbitrarily large n. We show that this happens for a direct
sum of Pedersen-Petersen C*-algebras, and analyze some other interesting
properties of these C*-algebras.
