  Let $\Gamma$ be a discrete group. To every ideal in $\ell^{\infty}(\G)$ we
associate a C$^*$-algebra completion of the group ring that encapsulates the
unitary representations with matrix coefficients belonging to the ideal. The
general framework we develop unifies some classical results and leads to new
insights. For example, we give the first C$^*$-algebraic characterization of
a-T-menability; a new characterization of property (T); new examples of
"exotic" quantum groups; and, after extending our construction to
transformation groupoids, we improve and simplify a recent result of Douglas
and Nowak.
