Vol. 96, No. 2, 1981

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Invariant subspaces of nonselfadjoint crossed products

Michael John McAsey

Vol. 96 (1981), No. 2, 457–473

Let L be the von Neumann algebra crossed product determined by a maximal abelian selfadjoint algebra L(X) and an automorphism of L(X). The algebra L is generated by a bilateral shift Lδ and an abelian algebra ML isomorphic to L; the non-selfadjoint subalgebra L+ of L is defined to be the weakly closed algebra generated by Lδ and ML. The commutant of L is the algebra R, also a crossed product. The invariant subspace structure of L+ is investigated. It is shown that full, pure invariant subspaces for L+ are unitarily equivalent by a unitary operator in R if and only if their associated projections are equivalent in ML. Furthermore, a multiplicity function can be associated with each invariant subspace. The algebra R contains a subalgebra R+ analogous to L+. It is shown that subspaces invariant for both algebras L+ and R+ can be parameterized in terms of certain subsets of the cartesian product Z × X.

Mathematical Subject Classification 2000
Primary: 46L55
Secondary: 47A15
Received: 19 January 1979
Revised: 1 April 1980
Published: 1 October 1981
Michael John McAsey