Contact Info
Pure MathematicsUniversity of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada
N2L 3G1
Departmental office: MC 5304
Phone: 519 888 4567 x43484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
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
Departmental office: MC 5304
Phone: 519 888 4567 x43484
Fax: 519 725 0160
Email: puremath@uwaterloo.ca
The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.