  We consider the class of groups called identity excluding which has the
property that any non-trivial irreducible unitary representation restricted to
a dense subgroup does not weakly contain the trivial representation. For
adapted and strictly aperiodic probability measures on these groups, it is
known that the averages of unitary representations converge strongly. We show
that motion group of a totally disconnected nilpotent group and certain class
of p-adic algebraic groups which includes gruops whose solvable radical is type
R are identity excluding. We also prove the convergence of averages of unitary
representations for split solvable algebraic groups, which are not necessarily
identity excluding.
