Weakly closed maximal triangular algebras are hyperreducible

Abstract

In this note we observe that the assertion in the title follows easily from known results. Recall that an algebra 6R of bounded linear operators on a Hilbert space is called [1] triangular if (R 6* is a maximal abelian von Neumann algebra. A triangular algebra is maximal if it is not properly contained in another triangular algebra; it is hyperreducible if the projections onto its invariant subspaces generate a maximal abelian von Neumann algebra. Now the theorem stated in the title can be proven as follows. Let a1 be a weakly closed maximal triangular algebra. Since aR is maximal the lattice of invariant subspaces of 61 is totally ordered [1, Lemma 2.3.3]. It follows from Theorem 2 of [2] that 61 is refexive, i.e., 61 contains every operator which leaves invariant every invariant subspace of (R. Hence, by Lemma 5.1 of [3], 61 is hyperreducible. This result shows one reason that hyperreducible maximal triangular algebras are nice (see [1]). Note that the proof of Lemma 5.1 of [3] shows that every reflexive triangular algebra (not just those with totally ordered invariant subspace lattices), is hyperreducible. Thus it follows (by a proof similar to the above) from Theorem 3 of [2] that a weakly closed triangular algebra whose lattice of invariant subspaces is the direct product of a family of totally ordered lattices is hyperreducible. In view of these results it seems reasonable to conjecture that every weakly closed triangular algebra is hyperreducible. A proof of this conjecture would be a partial answer to question (i) of [2].