Analysis Seminar

Nico Spronk, Pure Mathematics, University of Waterloo

"On similarity for completely bounded representations of Fourier algebras"

Let $G$ be a locally compact group.   Dixmier's unitarization theorem for
bounded continuous group representations may be restated as follows:  if
$G$ is amenable, then every bounded representation for the group algebra on a Hilbert space, $\pi:L^1(G)\to\mathcal{B}(\mathcal{H})$, admits an invertible $S$ in $\mathcal{B}(\mathcal{H})$ for which
\[
S\pi(\cdot)S^{-1}\text{ is a }*\text{-representation, and
}\|S\|\|S^{-1}\|\leq \|\pi\|^2. \tag{$\dagger$}
\]
In the `90s, Pisier showed that ($\dagger$) implies amenability of $G$.
The Fourier algebra $A(G)$ is the dual object to $L^1(G)$ in a manner
which generalizes Pontryagin duality.  It is a commutative self-adjoint
Banach algebra of functions on $G$ which is the predual of the von Neuman algebra generated by the left regular representation of $G$.  As such, the operator space structure on $A(G)$ is generally non-trivial.  However, every $*$-representation of $A(G)$ factors through the commutative C*-algebra of continuous functions vanishing at infinity
$\mathcal{C}_0(G)$, and hence is completely bounded.  Due to the
considerations around the duality of $A(G)$ with $L^1(G)$, we suspect that for any completely bounded representation
$\pi:A(G)\to\mathcal{B}(\mathcal{H})$ that there is an $S$ in
$\mathcal{B}(\mathcal{H})$ for which an analogue of ($\dagger$) holds.
H.H. Lee (Seoul) and E. Samei (Saskatchewan) and I have found a proof for this result for a wide class of groups which includes amenable groups and small-invariant neighbourhood (hence discrete) groups.

MC 5417