Reflexive algebras with finite width lattices: Tensor products, cohomology, compact perturbations

Abstract

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ∗-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg( L 1 ⊗ L 2) = Alg L 1 ⊗ Alg L 2, where each L i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each L i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if L is a finite-width commutative subspace lattice and K is the set of compact operators then the quasitriangular algebra Alg L + K is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.