Invariant subspaces and cocyles in nonselfadjoint crossed products
This paper is a generalization of two earlier joint works by the present author, \textit{K.-S. Saito}, and \textit{M. McAsay} and \textit{P. S. Muhlu} [Trans. Am. Math. Soc. 248, 381-409 (1979; Zbl 0372.46068) and Math. Soc. Jap. 33, 485-495 (1981; Zbl 0469.46046)]. Let \({\mathcal L}\) be the von Neumann algebra crossed product of M, a finite von Neumann algebra, and \(\{\alpha_{\gamma}\}_{\gamma \in \Gamma}\), a one-parameter group of trace preserving *-automorphisms of \({\mathcal M}\), where \(\Gamma\) is a discrete archimedean totally ordered abelian group. Let G be the dual group of \(\Gamma\). Let \({\mathcal L}_+\) be the subalgebra \(\{\) \(x\in {\mathcal L}:\) spectrum of x (with respect to dual automorphism group G on \({\mathcal L})\) is non-negativ\(e\}\). One of the main results of this paper is: Theorem (6.3). If \({\mathcal M}\) is a factor, then \({\mathcal L}_+\) is a maximal \(\sigma\)-weakly closed subalgebra of \({\mathcal L}.\) This result appears as a consequence of a study of invariant subspaces for \({\mathcal L}_+\) as well as a study of the (unitary) cocycles of so called ''pure, full, left normalized'' invariant subspaces of \(L^ 2\).
one-parameter group of trace preserving *-automorphisms
scientific article; zbMATH DE number 3851808
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