Nest algebras and similarity transformations (Q1084648)

Answering an old question of J. R. Ringrose, the author shows that similar continuous nests on a separable Hilbert space can fail to be unitarily equivalent. Moreover, arbitrary continuous nests are similar and every maximal nest is similar to a multiplicity-one nest. Then the author considers non-commutative nests. It is proved among other things that a complete nest \({\mathcal N}\) is countable if and only if every positive invertible operator T admits a factorization \(T=A^*A\) for some positive A in (alg \({\mathcal N})\cap (alg {\mathcal N})^{-1}\). If \({\mathcal N}\) is an uncountable nest generating an atomic von Neumann algebra then \({\mathcal N}\) is similar to a nest with non-trivial continuous part.