Skip to main content

Free probability setting

  • 1656 Accesses

Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 1957)

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alice Guionnet .

Rights and permissions

Reprints and Permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

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

Download citation