von Neumann algebras generated by operators similar to normal operators

Warren R. Wogen

Vol. 37 (1971), No. 2, 539–543

DOI: 10.2140/pjm.1971.37.539

Abstract

A normal operator generates an abelian von Neumann algebra. However, an operator which is similar to a normal operator may generate a von Neumann algebra which is not even type I. In fact, it is shown that if 𝒜 is a von Neumann algebra on a separable Hilbert space and 𝒜 has no type II finite summand, then 𝒜 has a generator which is similar to a self-adjoint and 𝒜 has a generator which is similar to a unitary. The restriction that 𝒜 have no type II finite summand can be removed provided that it is assumed that every type II finite von Neumann algebra has a single generator.

Mathematical Subject Classification 2000

Primary: 46L10

Secondary: 47B20

Milestones

Received: 23 October 1970

Published: 1 May 1971