Traces, Ultrapowers and the Pedersen-petersen C*-algebras

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 22 א0 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. Let A be a C*-algebra and let U be an ultrafilter on N. Every sequence τn, for n ∈ N, of traces on A defines a trace limn→U τn on the ultrapower AU (see §1 for definitions). We say that such traces on AU are trivial. Note that T (A) is in duality with AU if we extend τ ∈ T (A) to AU by τ((an)) = limn→U τ(an) (here (an) ∈ l∞(A) is a representing sequence of an element of AU ), and we have the following diagram.