Logo image
Back
Nonselfadjoint crossed products (Invariant subspaces and maximality)
Journal article   Open access   Peer reviewed

Nonselfadjoint crossed products (Invariant subspaces and maximality)

Michael McAsey, Paul S. Muhly and Kichi Suke Saito
Transactions of the American Mathematical Society, Vol.248(2), pp.381-409
03/01/1979
DOI: 10.1090/S0002-9947-1979-0522266-3
url
https://doi.org/10.1090/S0002-9947-1979-0522266-3View
Published (Version of record) Open Access

Abstract

Let (formula present) be the von Neumann algebra crossed product determined by a finite von Neumann algebra M and a trace preserving automorphism. In this paper we investigate the invariant subspace structure of the subalgebra (formula present) + of (formula present) consisting of those operators whose spectrum with respect to the dual automorphism group on (formula present) is nonnegative, and we determine conditions under which (formula present)+ is maximal among the o-weakly closed subalgebras of (formula present). Our main result asserts that the following statements are equivalent: (1) M is a factor; (2) (formula present)+ is a maximal a-weakly closed subalgebra of (formula present); and (3) a version of the Beurling, Lax, Halmos theorem is valid for (formula present)+. In addition, we prove that if (formula present) is a subdiagonal algebra in a von Neumann algebra (formula present) and if a form of the Beurling, Lax, Halmos theorem holds for (formula present), then (formula present) is isomorphic to a crossed product of the form (formula present) and (formula present) is isomorphic to (formula present)+.
Crossed products Maximality questions Subdiagonal algebras Von Neumann algebras

Details

Metrics

17 Record Views
73 Times Cited - Web of Science
Logo image