The parabolic algebra revisited

Kastis, Eleftherios and Power, Stephen (2024) The parabolic algebra revisited. Israel Journal of Mathematics, 259 (2). pp. 559-587. ISSN 0021-2172

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Abstract

The parabolic algebra Ap is the weakly closed operator algebra on L2(ℝ) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions eiλx, λ≥0. It is reflexive, with an invariant subspace lattice LatAp which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of LatAp 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 LatAp is not synthetic relative to the H∞(ℝ) subalgebra of Ap. Also, various new operator algebras, derived from isometric representations and from compact perturbations of Ap, are defined and identified.