On operator algebras and operator ranges (Q1087816)

Some further results are given and some questions raised concerning operator algebras of countable range multiplicity. An example is given of an abelian range cyclic algebra that does not have finite strict multiplicity. On the other hand it is shown that every uniformly closed abelian range cyclic algebra is strictly cyclic. It is not known if the same conclusion holds for non-abelian algebras, even strongly closed ones. Additionally, it is shown that the commutant of a range cyclic algebra has a non-trivial invariant subspace. Nests of invariant subspaces of algebras of finite range multiplicity are also discussed.