Exponential Equations Related to the Quantum ?ax + b? Group

Abstract

We study pairs b,β of unbounded selfadjoint operators, satisfying commutation rules inspired by the quantum ‘ax+b’ group [19]: bβ=−βb and β2=id except for kerb, on which β2=0. We find all measurable, unitary-operator valued functions F satisfying the exponential equation: F(b, β)F(d, δ)=F((b, β) (d, δ)), where d, δ satisfy the same commutation rules as b, β, and is modeled after the comultiplication of the quantum ‘ax+b’ group. This result is crucial for classification of all unitary representations of the quantum ‘ax+b’ group, which is achieved in our forthcoming paper [12].