The parabolic algebra revisited
Abstract
The parabolic algebra Ap is the weakly closed algebra on L2(R) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions eiλ x, λ ≥ 0. This algebra is reflexive with an invariant subspace lattice, Lat Ap, 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 Ap is nonsynthetic relative to the maximal abelian multiplication subalgebra of Ap. Also, operator algebras derived from isometric representations of Ap and from compact perturbations are defined and determined.