Can you compute the operator norm?

In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on a Hilbert space. We show that the operator norm in the universal unitary representation is computable if the group is residually finite-dimensional or amenable with a decidable word problem. Moreover, we relate the computability of the operator norm on the group F₂ × F₂ to Kirchberg’s QWEP Conjecture, a fundamental open problem in the theory of operator algebras.