On maximal non-selfadjoint reflexive algebras associated with a double triangle lattice (Q904649)

Let \(\mathcal H\) be a Hilbert space of dimension greater than two and let \(\mathcal B(\mathcal H)\) be the algebra of all bounded linear operators on \(\mathcal H\). Let \(\mathcal L\) be a lattice in \(\mathcal B(\mathcal H)\) and \(\mathrm{Alg }\mathcal L\) the lattice algebra. \(\mathcal L\) is called a double triangle lattice if \(\mathcal L=\{P_i:i=1,2,3\}\) if \(P_i\wedge P_j=0\) as well as \(P_i\vee P_j=I\) for any \(i\not=j\). The authors prove that, if the double triangle lattice \(\mathcal L\) generates a \(II_1\) factor von Neumann algebra, then the lattice algebra \(\mathrm{Alg }\mathcal L\) is maximal non-selfadjoint in the class of all weak operator topology closed subalgebras with the same diagonal subalgebra \(\mathrm{Alg }\mathcal L\cap \mathrm{Alg }\mathcal L^*\).