Abstract
The parabolic algebra \({{\cal A}_p}\) is the weakly closed operator algebra on \({L^2}(\mathbb{R})\) generated by the unitary semigroup of right translations and the unitary semigroup of multiplication by the analytic exponential functions \({e^{i\lambda x}},\lambda\ge 0\). It is reflexive, with an invariant subspace lattice \({\rm{Lat}}{{\cal A}_p}\) which is naturally homeomorphic to the unit disc (Katavolos and Power, 1997). The structure of \({\rm{Lat}}{{\cal A}_p}\) 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 \({\rm{Lat}}{{\cal A}_p}\) is not synthetic relative to the \({H^\infty}(\mathbb{R})\) subalgebra of \({{\cal A}_p}\). Also, various new operator algebras, derived from isometric representations and from compact perturbations of \({{\cal A}_p}\), are defined and identified.
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This work was supported by the Engineering and Physical Sciences Research Council [EP/P01108X/1] and by an LMS Early Career Fellowship [ECF-1920-66]
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Kastis, E., Power, S.C. The parabolic algebra revisited. Isr. J. Math. 259, 559–587 (2024). https://doi.org/10.1007/s11856-023-2550-4
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DOI: https://doi.org/10.1007/s11856-023-2550-4