We consider traces on the reduced and full C*-algebras, $C^*_r(G)$ and $C^*(G)$, of a locally compact group $G$. These traces can be understood very well for compactly generated $G$, in particular connected and almost connected $G$. We have for compactly generated $G$ an alternate proof of the result of Kennedy and Raum that $C^*_r(G)$ admits a trace exactly when $G$ has an open amenable radical. We also have a simple proof of Ng’s result that $G$ is amenable exactly when $C^*_r(G)$ is nuclear and admits a trace. We can then exploit a recent result of Schafhauser on AF-embeddability to characterize for which second countable almost connected $G$, $C^*(G)$ is AF-embeddable, generalizing some results of Beltita and Beltita; and we show that any tracially separated amenable group admits quasidaigonal $C^*_r(G)$. This is joint work with B. Forrest (Waterloo) and M. Wiersma (Winnipeg).

In a series of recent papers, Ebrahimi-Fard and Patras developed an algebraic approach for cumulants in non-commutative probability based on a combinatorial Hopf algebra of words on words on an alphabet. In particular, they showed that the combinatorial moment-cumulants formulas, expressed in terms of non-crossing partitions, can be retrieved from specific fixed-point equations involving linear functionals on a Hopf algebra.

In this talk, we discuss a combinatorial formula for the antipode in this Hopf algebra, which is represented in terms of Schröder trees, which have recently appeared in the context of non-commutative probability theory. Finally, we will see the implications of the antipode formula in non-commutative probability, namely, cumulant-moment formulas and free Wick polynomials in terms of Schröder trees. Based on an ongoing joint work with Yannic Vargas.

Naor and Schechtman recently introduced the so-called metric Xp inequalities, an obstruction for embeddings of Lq into Lp whenever 2 < q < p . This invariant was refined by Naor via a fundamental inequality in the Hamming cube which strongly relies on Fourier analysis. In this talk, we will show that this latter result can be understood within the frame of noncommutative harmonic analysis. In particular, the case over the group von Neumann algebra of the free group will be described. Moreover, we will briefly discuss some metric consequences and possible further work. This is joint work with José M. Conde-Alonso and Javier Parcet.