## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Friday, October 2, 2015 — 3:30 PM EDT

**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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x33484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1