Brian Forrest, Department of Pure Mathematics, University of Waterloo
"Exotic Ideals in the Fourier-Stieltjes Algebra of a Locally Compact Group"
There are two well known $C^*$-algebras associated with any locally compact group $G$, the full group $C^*$-algebra $C^*(G)$ and the reduced $C^*$-algebra $C_r^*(G)$. It is well known that these two $C^*$-algebras coincide precisely when the group $G$ is amenable. Recently, there has been considerable interest in identifying natural intermediate $C^*$-algebras sitting strictly between these two canonical algebras for various non-amenable groups such as the free group $\mathbb{F_2}$ on two generators. One way to generate such \textit{exotic} $C^*$-algebras is to consider various $L_p$-representations of the underlying group. (A unitary representation $\pi$ of $G$ is said to be an $L_p$-representation if there is a dense set of vectors $\xi \in H_\pi$ such that the coefficient function $u(x)=<\pi(x)\xi,\xi> \in L^p(G)$). For example, building on work of Brown and Guentner, Okayasu showed that if $2\leq p<q<\infty$, then for $\mathbb{F_2}$ the $C^*$-algebras resulting from the universal $L^p$ and $L^q$ representations are distinct.
In this talk we will focus on the dual version of this problem. In particular, we show that for large classes of locally compact groups, including even abelian groups, that these $L^p$-representations generate distinct exotic ideals in the Fourier-Stieltjes of $G$ which contain the Fourier algebra. We also look at some basic properties of these exotic ideals that distinguish them from the Fourier algebra itself.
This is joint work with Zsolt Tanko and Matthew Wiersma.
MC 5417