Abstract
Let us notice that by definition, a von Neumann algebra contains only bounded operators. The theory nevertheless allows us to consider unbounded operators thanks to the notion of affiliated operators. A densely defined selfadjoint operator X on H is said to be affiliated to A iff for any Borel function f on the spectrum of X, f(X) ? A (see [167, p.164]). Here, f(X) is well defined for any operator X as the operator with the same eigenvectors as X and eigenvalues given by the image of those of X by the map f. Murray and von Neumann have proved that if X and Y are affiliated with A, aX + bY is also affiliated with A for any a, b ? C.
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© 2009 Springer-Verlag Berlin Heidelberg
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Guionnet, A. (2009). Free probability setting. In: Large Random Matrices: Lectures on Macroscopic Asymptotics. Lecture Notes in Mathematics(), vol 1957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69897-5_17
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DOI: https://doi.org/10.1007/978-3-540-69897-5_17
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