Kastis, Lefteris and Power, Stephen (2015) The operator algebra generated by the translation, dilation and multiplication semigroups. Journal of Functional Analysis, 269. pp. 3316-3335. ISSN 0022-1236
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Abstract
The weak operator topology closed operator algebra on $L^2(\bR)$ generated by the one-parameter semigroups for translation, dilation and multiplication by $e^{i\lambda x}, \lambda \geq 0,$ is shown to be a reflexive operator algebra, in the sense of Halmos, with invariant subspace lattice equal to a binest. This triple semigroup algebra, $\A_{ph}$, is antisymmetric in the sense that $\A_{ph}\cap \A_{ph}^*=\bC I$, it has a nonzero proper weakly closed ideal generated by the finite-rank operators, and its unitary automorphism group is $\bR$. Furthermore, the $8$ choices of semigroup triples provide $2$ unitary equivalence classes of operator algebras, with $\A_{ph}$ and $\A_{ph}^*$ being chiral representatives.