**
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