Non-selfadjoint operator algebras generated by unitary semigroups

Abstract

The parabolic algebra was introduced by Katavolos and Power, in 1997, as the weak^∗-closed operator algebra acting on L²(R) that is generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive, in the sense of Halmos, and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces in the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on L^p(R), where 1 < p < ∞. It is also shown that the reflexive closures of the Fourier binests on L^p(R) are all order isomorphic for 1 < p < ∞. The weakly closed operator algebra on L²(R) generated by the one-parameter semigroups for translation, dilation and multiplication by e^iλx, λ ≥ 0, is shown to be a reflexive operator algebra with invariant subspace lattice equal to a binest. This triple semigroup algebra, A_ph, is antisymmetric, it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is R. Furthermore, the 8 choices of semigroup triples provide 2 unitary equivalence classes of operator algebras, with A_ph and (A_ph)^∗ being chiral representatives. In chapter 4, we consider analogous operator norm closed semigroup algebras. Namely, we identify the norm closed parabolic algebra A_p with a semicrossed product for the action on analytic almost periodic functions by the semigroup of one-sided translations and we determine its isometric isomorphism group. Moreover, it is shown that the norm closed triple semigroup algebra Aph^G+ is the triple semi-crossed product A_p ×_v G₊, where v denotes the action of one-sided dilations. The structure of isometric automorphisms of A_ph^G+ is determined and A_ph^G+ is shown to be chiral with respect to isometric isomorphisms. Finally, we consider further results and state open questions. Namely, we show that the quasicompact algebra QA_p of the parabolic algebra is strictly larger than the algebra CI + K(H), and give a new proof of reflexivity of certain operator algebras,generated by the image of the left regular representation of the Heisenberg semigroup H₊.