A unitarizable group is a discrete group 
such that every uniformly
bounded representation 
on a complex Hilbert space
is similar to a unitary representation.
Uniform boundedness means 
. Similarity requires one bounded
invertible operator 
for which 
is unitary for every 
. The operator 
may depend on the representation.
For a finite group, averaging the inner product over the group makes any representation unitary in an equivalent inner product. The Dixmier unitarizability problem asks whether every unitarizable group is an amenable group. Strong Ulam stability instead concerns approximate multiplication by unitary transformations.
