Let $G$ be a locally compact group. There is a natural correspondence between continuous positive definite functions $u$ on $G$ with $u(e)=1$, and states on the universal C*-algebra $C^*(G)$. With this correspondence traces are those continuous positive definite functions $u$ for which $u(st)=u(ts)$ and $u(e)=1$. Then $N_u = \{ s \in G : u(s)=1 \}$ is a closed normal subgroup of $G$ and the intersection of such ``trace kernels” is the trace kernel $N_{Tr}$. I wish to say as much as I can about the structure of $G/N_{Tr}$, sometimes for certain classes of groups. I also wish to speak about reduced traces, i.e.\ traces on the reduced C*-algebra $C^*_r(G)$; and add non-discrete examples of groups with unique reduced trace. I will also discuss amenable traces and factorization property for some classes of locally compact groups.

This is joint work with Brian Forrest (Waterloo) and Matthew Wiersma (UC San Diego and Winnipeg).