The parabolic algebra revisited

The parabolic algebra A(p) is the weakly closed operator 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 >= 0. It is reflexive, with an invariant subspace lattice LatA(p) which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of LatA(p) is used to classify strongly irreducible isometric representations of the partial Weyl commutation relations. A formal generalisation of Arveson's notion of a synthetic commutative subspace lattice is given for general subspace lattices, and it is shown that LatA(p) is not synthetic relative to the H-infinity(R) subalgebra of A(p). Also, various new operator algebras, derived from isometric representations and from compact perturbations of A(p), are defined and identified.