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

http://dx.doi.org/10.4153/CMB-2006-012-3
Canad. Math. Bull. 49(2006), 117-126

  Published:2006-03-01
 Printed: Mar 2006

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 $SL_2(\R_+)$.