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A Double Triangle Operator Algebra From SL2(ℝ+)

Published online by Cambridge University Press:  20 November 2018

R. H. Levene
R. H. Levene*
Affiliation:
Pure Mathematics Department, University of Waterloo, Waterloo, ON, N2L 3G1 e-mail: rupert.levene@cantab.net
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Abstract

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We consider the ${{w}^{*}}$-closed operator algebra ${{\mathcal{A}}_{+}}$ generated by the image of the semigroup $S{{L}_{2}}\left( {{\mathbb{R}}_{+}} \right)$ under a unitary representation $\rho$ of $S{{L}_{2}}\left( \mathbb{R} \right)$ on the Hilbert space ${{L}^{2}}\left( \mathbb{R} \right)$. We show that ${{\mathcal{A}}_{+}}$ is a reflexive operator algebra and ${{\mathcal{A}}_{+}}=\text{Alg }\mathcal{D}$ where $\mathcal{D}$ is a double triangle subspace lattice. Surprisingly, ${{\mathcal{A}}_{+}}$ is also generated as a ${{w}^{*}}$-closed algebra by the image under $\rho$ of a strict subsemigroup of $S{{L}_{2}}\left( {{\mathbb{R}}_{+}} \right)$.

Keywords

46K5047L55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 49 , Issue 1 , 01 March 2006 , pp. 117 - 126
Copyright
Copyright © Canadian Mathematical Society 2006

Footnotes

The author is supported by an EPSRC grant.

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