# Definition of a uniformly bounded dual of a group

The unitary dual of a group $$G$$ is the set of equivalence classes of irreducible unitary representations of $$G$$ with the Fell topology. (This topology is defined using convergence of positive definite functions, or equivalently, diagonal matrix coefficients, associated to the representations).

Question: How to define a "uniformly bounded dual" of $$G$$ using uniformly bounded representations on a Hilbert space and what topology should it be equipped with?

Such a uniformly bounded dual is mentioned a few times in the literature, but usually vaguely defined to be "analogous to the unitary dual". In particular, there is hardly any explicit statement about what topology is used.

I am interested in this since I would like to understand a series of results of M. Cowling from the early 1980s on a stronger version of Kazhdan's property (T). Cowling proved that for a simple Lie group $$\Gamma$$, the trivial representation is isolated in such a uniformly bounded dual if and only if the rank of $$\Gamma$$ is $$\ge 2$$.

In the case of $$Sp(n,1)$$ Cowling constructs an explicit family of uniformly bounded representations that approximate the trivial representation. The best description of his construction I have found is in his abstract from an Oberwolfach workshop from 2001 on the geometrization of property (T) (see MFO report, the following abstract is on page 3):

• Note that for unitary representations it is usually easier to describe neighborhoods of the trivial representation than of an arbitrary unitary rep (there are subtleties with convex combinations...). Here you might want to specify how you expect the operator norm to converge. If a sequence $\pi_i$ converges to the identity, and $K$ is a compact neighbourhood of 1 in the group, do you want $\sup_K\|\pi_i(g)\|$ to tend to 1? or just to be bounded? Or none? – YCor Apr 7 '13 at 18:19
• Yves, no behavior of norms of the representations is assumed. In fact Cowling has proved that for $Sp(n,1)$ the trivial representation can be approximated by uniformly bounded ones, even though $Sp(n,1)$ has (T). However, the norms of these uniformly bounded representations blow up to infinity as one approaches the trivial one. – user2412 Apr 7 '13 at 18:26
• Are you allowing u.b. representations on (reflexive) Banach spaces other than Hilbert space? – Yemon Choi Apr 7 '13 at 22:19
• Yemon, I am only considering the Hilbert space. – user2412 Apr 8 '13 at 4:07
• Thanks for the suggestion. I have tried the obvious alternate routes already, moreover, it seemed that MO is the natural place to ask such a question. Several other people also reference or use Cowling's results in their work. For instance, Julg has an approach to the Baum-Connes conjecture with coefficients for $Sp(n,1)$ and mentioned the "u.b. dual" explicitly; Shalom's unpublished proof that $Sp(n,1)$ admits a proper affine action with uniformly bounded linear part is also based on Cowling's result, as far as I know. However, the meaning of the term "isolated" is never defined precisely. – user2412 Apr 10 '13 at 8:15