Equivalence classes of invariant subspaces in nonselfadjoint crossed products (Q1067562)

The Beurling-Lax-Halmos theorem on invariant subspaces of the shift may be viewed as showing that \(H^ 2\) is the canonical invariant subspace for the shift; all other invariant subspaces are ''multiples'' of \(H^ 2\) (i.e., \(\phi H^ 2\) for \(\phi\) an inner function). This theme is continued in the present paper. The shift operator is replaced by a non- selfadjoint subalgebra, \({\mathcal L}_+\), of a crossed product, \({\mathcal L}\) (and so \({\mathcal L}_+\) contains a shift operator). Rather than a single canonical invariant subspace, the replacement for \(H^ 2\) is a perspicuous collection of subspaces which will serve as canonical models for all \({\mathcal L}_+\)-invariant subspaces, in the following sense. The family \(\{\) \({\mathcal M}_ i\}_{i\in I}\) of \({\mathcal L}_+\)-full, pure, invariant subspaces is a set of canonical models in case i) \(P_{{\mathcal M}_ i}\) is not unitarily equivalent to \(P_{{\mathcal M}_ j}\) by a unitary operator in \({\mathcal L}'\) and ii) for every \({\mathcal L}_+\)-invariant subspace \({\mathcal M}\), there is an \({\mathcal M}_ i\) so that \({\mathcal M}=V{\mathcal M}_ i\), for a partial isometry V in \({\mathcal L}'.\) In this paper, the crossed product is constructed from the von Neumann algebra \(M=L^{\infty}(X,\mu)\otimes N\) (X a standard Borel space with finite positive measure \(\mu\), N a finite factor) and automorphism \(\alpha ={\tilde \tau}\otimes \alpha_ 0\) (\({\tilde \tau}\) is a *- automorphism of \(L^{\infty}(X,\mu)\) induced by an ergodic, invertible, measure preserving transformation; \(\alpha_ 0\) is a *-automorphism of N). Associated with each nonnegative measurable function m on X satisfying \(\int m(t)d\mu (t)\leq \mu (X)\), there exists a full invariant subspace \({\mathcal M}\). These multiplicity functions m are used to index the set of canonical models for all the subspaces invariant for \({\mathcal L}_+\). For additional results on this theme see \textit{B. Solel}'s paper in Trans. Am. Math. Soc. 279, 825-840 (1983; Zbl 0556.46037) and \textit{S. Kawamura} and \textit{Tomimori} in Bull. Yamagata Univ. 11, 17-31 (1984).