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A Double Triangle Operator Algebra From SL2(ℝ+)
Published online by Cambridge University Press: 20 November 2018
Abstract
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
$S{{L}_{2}}\left( {{\mathbb{R}}_{+}} \right)$.
- Type
- Research Article
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- Copyright
- Copyright © Canadian Mathematical Society 2006
Footnotes
The author is supported by an EPSRC grant.
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