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Transactions of the American Mathematical Society

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The invariant subspace structure of nonselfadjoint crossed products

Author: Baruch Solel
Journal: Trans. Amer. Math. Soc. 279 (1983), 825-840
MSC: Primary 46L55; Secondary 46L10
MathSciNet review: 709586
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Abstract: Let $ \mathcal{L}$ be the von Neumann algebra crossed product determined by a finite von Neumann algebra $ M$ and a trace preserving $ ^{\ast}$-automorphism $ \alpha $ of $ M$. We study the invariant subspace structure of the subalgebra $ {\mathcal{L}_ + }$ of $ \mathcal{L}$ consisting of those operators whose spectrum with respect to the dual automorphism group on $ \mathcal{L}$ is nonnegative. We investigate the conditions for two invariant subspaces $ {\mathcal{M}_1}$, and $ {\mathcal{M}_2}$ (with $ {Q_{1}},{Q_2}$ the corresponding orthogonal projections) to satisfy $ {Q_1} = {R_\upsilon }\,{Q_2}\,R_\upsilon^{\ast}$ for some partial isometry $ {R_{\upsilon }}$ in $ \mathcal{L}^{\prime}$. We use this analysis to find the general form of a $ \sigma $-weakly closed subalgebra of $ \mathcal{L}$ that contains $ {\mathcal{L}_ + }$.

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Keywords: Crossed products, invariant subspace, nonselfadjoint algebra, shift operator, centre-valued trace
Article copyright: © Copyright 1983 American Mathematical Society

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