Spectral Subspaces of Automorphism Groups of Type I Factors

Abstract

A well known theorem of M. H. Stone published in 1930 [5] establishes a one-to-one correspondence between strongly continuous unitary representations of the real line in Hilbert space and spectral resolutions in that space. More recently, W. B. Arveson [1] generalized the notion of spectral subspace (corresponding to spectral projection in the Hilbert space situation) to a large class of representations in Banach space. His theory is particularly useful in the case of an ultraweakly continuous one-parameter group of *-automorphisms a = {a,} of a von Neumann algebra M. In fact, by [1, Lemma 2, p. 233], the spectral subspaces of M of the form M[X, +00) completely determine the representation a. In view of this result it is natural to ask how to synthesize a from the M[A, +oo). Although a constructive answer to this question seems hard to give in general, it can quite easily be provided in the special case where M is a type I factor. The solution (Lemma l(b) below) is based on an idea of E. Stormer [6] and actually makes use of Stone's theorem in the Hilbert space of Hilbert-Schmidt operators associated with M. Closely related to the synthesis problem there is also a characterization problem: given a family {M;};eR of ultraweakly closed subspaces of a von Neumann algebra M, when does there exist a one-parameter group a of *-automorphisms of M such that M-,, = M\l, oo) for all >, e Ul A number of necessary conditions come immediately to mind: