Certain invariant subspace structure of analytic crossed products (Q1101651)

In this note, we study the invariant subspace structure of analytic crossed products in the following setting. Let X be a standard Borel space with a \(\sigma\)-finite infinite positive measure \(\mu\), that is, \(\mu (X)=\infty\). Let \(\tau\) be an invertible measure preserving ergodic transformation on X. We consider the analytic crossed product \({\mathcal L}_+\) determined by \(L^{\infty}(X,\mu)\) and \(\tau\). Then we study the complete set of canonical models for Lat(\({\mathcal L}_+)\) which is the set of all left-pure left-invariant subspaces. Then we prove that there exists a left-pure, left-full, left-invariant subspace \({\mathcal M}_{\infty}\) such that, for every \({\mathcal M}\in Lat({\mathcal L}_+)\), there is a partial isometry V in \({\mathcal R}\) satisfying \({\mathcal M}_{\infty}=V{\mathcal M}\). If \(\mu\) is a finite measure, then we refer to \textit{M. AcAsey} [Pac. J. Math. 96, 457-473 (1981; Zbl 0424.46044)] and \textit{B. Solel} [ibid. 113, 201-214 (1984; Zbl 0586.46051)].