The multiplicity functions of invariant subspaces for non selfadjoint crossed products (Q1071274)

Let \({\mathcal L}\) be the von Neumann algebra crossed product determined by a maximal selfadjoint algebra \(L^{\infty}(X)\) and an ergodic automorphism of \(L^{\infty}(X)\). The algebra is generated by a bilateral shift L and an abelian algebra \(M_ L\) isomorphic to \(L^{\infty}(X)\). The non selfadjoint subalgebra \({\mathcal L}_+\) of \({\mathcal L}\) is the weakly closed algebra generated by L and \(M_ L.\) In this paper the invariant subspaces of \({\mathcal L}_+\) are studied using the notion of a mutliplicity function. A necessary and sufficient condition for a function to be a multiplicity function is given. Some of the results of this paper were later extended to more general settings [see \textit{B. Solel}, Trans. Am. Math. Soc. 279, 825--840 (1983; Zbl 0556.46037); and J. Funct. Anal. 58, 1--19 (1984; Zbl 0587.46055)].