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 M′L. 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.
