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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Nonselfadjoint crossed products (invariant subspaces and maximality)


Authors: Michael McAsey, Paul S. Muhly and Kichi-Suke Saito
Journal: Trans. Amer. Math. Soc. 248 (1979), 381-409
MSC: Primary 46L10
MathSciNet review: 522266
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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 automorphism. In this paper we investigate 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, and we determine conditions under which $ {\mathcal{L}_ + }$ is maximal among the $ \sigma $-weakly closed subalgebras of $ \mathcal{L}$. Our main result asserts that the following statements are equivalent: (1) M is a factor; (2) $ {\mathcal{L}_ + }$ is a maximal $ \sigma $-weakly closed subalgebra of $ \mathcal{L}$; and (3) a version of the Beurling, Lax, Halmos theorem is valid for $ {\mathcal{L}_ + }$. In addition, we prove that if $ \mathfrak{A}$ is a subdiagonal algebra in a von Neumann algebra $ \mathcal{B}$ and if a form of the Beurling, Lax, Halmos theorem holds for $ \mathfrak{A}$, then $ \mathcal{B}$ is isomorphic to a crossed product of the form $ \mathcal{L}$ and $ \mathfrak{A}$ is isomorphic to $ {\mathcal{L}_ + }$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1979-0522266-3
PII: S 0002-9947(1979)0522266-3
Keywords: Crossed products, von Neumann algebras, subdiagonal algebras, maximality questions
Article copyright: © Copyright 1979 American Mathematical Society