Levene, Rupert Howard
(2004)
*Lie Semigroup Operator Algebras.*
PhD thesis, Lancaster University.

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## Abstract

The parabolic algebra Ap and the hyperbolic algebra Ah are nonselfadjoint w*-closed operator algebras which were first considered by A. Katavolos and S. C. Power. In [KP97] and [KP02] they showed that their invariant subspace lattices are homeomorphic to compact connected Euclidean manifolds, and that the parabolic algebra is reflexive in the sense of Halmos. We give a new proof of the reflexivity of the parabolic algebra through analysis of Hilbert-Schmidt operators. We also show that there are operators in Ap with nontrivial kernel. We then consider some natural "companion algebras" of the parabolic algebra which leads to a compact subspace lattice known as the Fourier-Plancherel sphere. We show that the unitary automorphism group of this lattice is isomorphic to a semidirect product of R2 and SL2(R). A proof that the hyperbolic algebra is reflexive follows by an essentially identical analysis of Hilbert-Schmidt operators to that which was used to establish the reflexivity of Ap. We also present a transparent proof of a known result concerning a strong operator topology limit of projections. Both of the Katavolos-Power algebras are generated as w*-closed operator algebras by the image of a semigroup of a Lie group under a unitary-valued representation. Following [KP02], we call such operator algebras Lie semigroup operator algebras. We seek new examples of such algebras by considering the images of the semigroup SL2(R+) of the Lie group SL2(R) under unitary-valued representations of SL2 (R). We show that a particular Lie semigroup operator algebra A+ arising in this way is reflexive and that it is the operator algebra leaving a double triangle subspace lattice invariant. Surprisingly, A+ is generated as a w*-closed algebra by the image of a proper subsemigroup of SL2(R+).