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On the geometry of conditional expectations treated as projections on the L2-space

  • Andrzej Komisarski EMAIL logo and Adam Paszkiewicz
Published/Copyright: November 3, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 4

Abstract

We study the relative position of two measure-theoretical conditional expectations treated as orthogonal projections on the L2-space. We show that any pair of orthogonal projections P and Q on a separable Hilbert space with dim(P ∧ Q) = ∞ is unitarily equivalent to the conditional expectations under two sub-σ-fields of some probability space.

MSC 2010: Primary 46C07; 60A10; 47C15
Keywords: conditional expectation; L2-space; orthogonal projection

(Communicated by Sylvia Pulmannová)

References

[1] Conway, J. B.: A Course in Functional Analysis. Grad. Texts in Math. 96, Springer-Verlag, New York, 1990.Search in Google Scholar

[2] Davis, C.: Separation of two linear subspaces, Acta Sci. Math. (Szeged) 19 (1958), 172–187.Search in Google Scholar

[3] Dixmier, J.: Étude sur les variétés et les opérateurs de Julia, avec quelques applications, Bull. Soc. Math. France 77 (1949), 11–101.10.24033/bsmf.1403Search in Google Scholar

[4] Halmos, P. R.: Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389.10.1007/978-1-4613-8208-9_26Search in Google Scholar

[5] Takesaki, M.: Theory of Operator Algebras I, Springer-Verlag, Berlin, 2002.10.1007/978-3-662-10453-8Search in Google Scholar

Received: 2012-11-26
Accepted: 2014-1-20
Published Online: 2016-11-3
Published in Print: 2016-8-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0195
Keywords for this article
conditional expectation; L2-space; orthogonal projection

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