Can you compute the operator norm?

TL;DR

This paper explores the computability of the operator norm in unitary group representations, linking it to group properties and a major conjecture in operator algebra theory.

Contribution

It establishes conditions under which the operator norm is computable and connects this problem to Kirchberg's QWEP Conjecture.

Findings

Operator norm is computable for residually finite-dimensional groups.

Operator norm is computable for amenable groups with decidable word problem.

Computability of the operator norm relates to Kirchberg's QWEP Conjecture.

Abstract

In this note we address various algorithmic problems that arise in the computation of the operator norm in unitary representations of a group on 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 decidable word problem. Moreover, we relate the computability of the operator norm on the product of non-abelian free groups to Kirchberg's QWEP Conjecture, a fundamental open problem in the theory of operator algebras.