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A Double Triangle Operator Algebra From $SL_2(\R)$

Open Access article
 Printed: Mar 2006
  • R. H. Levene
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We consider the w$^*$-closed operator algebra $\cA_+$ generated by the image of the semigroup $SL_2(\R_+)$ under a unitary representation $\rho$ of $SL_2(\R)$ on the Hilbert~space $L_2(\R)$. We show that $\cA_+$ is a reflexive operator algebra and $\cA_+=\Alg\cD$ where $\cD$ is a double triangle subspace lattice. Surprisingly, $\cA_+$ is also generated as a w$^*$-closed algebra by the image under $\rho$ of a strict subsemigroup of $SL_2(\R_+)$.
MSC Classifications: 46K50, 47L55 show english descriptions Nonselfadjoint (sub)algebras in algebras with involution
Representations of (nonselfadjoint) operator algebras
46K50 - Nonselfadjoint (sub)algebras in algebras with involution
47L55 - Representations of (nonselfadjoint) operator algebras

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