A Double Triangle Operator Algebra From SL 2(R+)

Levene, R. H.

Canadian mathematical bulletin, Tome 49 (2006) no. 1, pp. 117-126

Résumé

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 $S{{L}_{2}}\left( {{\mathbb{R}}_{+}} \right)$ .

Détail

DOI : 10.4153/CMB-2006-012-3

Mots-clés : 46K50, 47L55