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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Analytic operator algebras (factorization and an expectation)


Author: Baruch Solel
Journal: Trans. Amer. Math. Soc. 287 (1985), 799-817
MSC: Primary 47D25; Secondary 46L99
DOI: https://doi.org/10.1090/S0002-9947-1985-0768742-2
MathSciNet review: 768742
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Abstract: Let $ M$ be a $ \sigma $-finite von Neumann algebra and $ {\{ {\alpha _t}\} _{t \in T}}$ a periodic flow on $ M$. The algebra of analytic operators in $ M$ is $ \{ a \in M:{\text{sp}_\alpha }(a) \subseteq {{\mathbf{Z}}_ + }\} $ and is denoted $ {H^\infty }(\alpha )$. We prove that every invertible operator $ a \in {H^\infty }(\alpha )$ can be written as $ a = ub$, where $ u$ is unitary in $ M$ and $ b \in {H^\infty }(\alpha ) \cap {H^\infty }{(\alpha )^{ - 1}}$. We also prove inner-outer factorization results for $ a \in {H^\infty }(\alpha )$.

Another result represents $ {H^\infty }(\alpha )$ as the image of a certain nest subalgebra (of a von Neumann algebra that contains $ M$) via a conditional expectation. As corollaries we prove a distance formula and an interpolation result for the case where $ M$ is an injective von Neumann algebra.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0768742-2
Keywords: Flow, analytic operators with respect to a flow, inner-outer factorization, expectation, distance estimate, interpolation
Article copyright: © Copyright 1985 American Mathematical Society