Traces, ultrapowers and the pedersen-petersen C∗-algebras

Abstract

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 22ℵ0 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.