A Note on the Similarity Theorem for Nest Algebras

In [4], nests were classified up to similarity simply in terms of order type and dimension. However the proof given there and exposited in [6] appears to rely on first classifying the compact perturbations of the nest algebras. This, in turn, required reliance on some difficult results on derivations of C*-algebras and in particular on the theorem of Johnson and Parrott [7]. However a careful examination of the proofs shows that this circuitous route can be avoided. This note is a study guide to allow the reader to wend his or her way through those necessary parts of the proof and avoid the unneeded results on quasitriangular operators if they choose. Historically, Andersen [1] showed that any two continuous nests were approximately unitarily equivalent; and used this to show that their quasitriangular operator algebras were unitarily equivalent. Larson [8] used this result to show that any two continuous nests were similar (although he was not able to control the induced order isomorphism). Then the author [4] classified all nests up to similarity, but took a route that passed first through a classification of quasitriangular operators. This mindset persisted in the treatment in our text [6]. A nest N is a complete chain of subspaces of a Hilbert space containing {0} and H. For this note, all Hilbert spaces will be separable. The nest algebra T (N ) consists of all operators leaving each element ofN invariant— the set of operators with this prescribed upper triangular form. We let PN denote the orthogonal projection of H onto N . Then we obtain