I will first talk about correlation matrices arising from mixed moments, in terms of traces, of unitaries, respectively, projections, and how these relate to Tsirelson's conjecture and the Connes Embedding Problem. I will then explain how the so-called factorizable maps can be parameterized by traces, which naturally leads to questions about different classes of traces on residually finite dimensional C*-algebras. Finally, I will talk about traces on products of C*-algebras vs products of traces on C*-algebras (after Ozawa).
The first two parts of the talk are joint work with Magdalena Musat, and the last part is a report on ongoing work with my current PhD student Henning Milhoej.