The operator algebra generated by the translation, dilation and multiplication semigroups

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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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.

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Journal Article
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Journal of Functional Analysis
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This is the author’s version of a work that was accepted for publication in Journal of Functional Analysis. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Functional Analysis, 269, 2015 DOI: 10.1016/j.jfa.2015.08.005
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14 Aug 2015 10:12
Last Modified:
11 Jul 2020 05:05