The parabolic algebra revisited

TL;DR

This paper revisits the parabolic algebra A_p, exploring its structure, invariant subspace lattice, and representations, revealing new insights into its non-synthetic nature and related operator algebras.

Contribution

It classifies strongly irreducible isometric representations of the partial Weyl relations and extends the concept of synthetic subspace lattices to noncommutative settings.

Findings

Lat A_p is homeomorphic to the unit disc

Lat A_p is nonsynthetic relative to a maximal abelian subalgebra

Operator algebras from isometric representations and perturbations are characterized

Abstract

The parabolic algebra A_p is the weakly closed algebra on L^2(R) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions e^{i\lambda x}, \lambda \geq 0. This algebra is reflexive with an invariant subspace lattice, Lat A_p, which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). This identification is used here to classify strongly irreducible isometric representations of the partial Weyl commutation relations. The notion of a synthetic subspace lattice is extended from commutative to noncommutative lattices and it is shown that Lat A_p is nonsynthetic relative to the maximal abelian multiplication subalgebra of A_p. Also, operator algebras derived from isometric representations of A_p and from compact perturbations are defined and determined.