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5 - Free probability

Published online by Cambridge University Press:  03 February 2011

Greg W. Anderson
Affiliation:
University of Minnesota
Alice Guionnet
Affiliation:
Ecole Normale Supérieure, Lyon
Ofer Zeitouni
Affiliation:
Weizmann Institute/University of Minnesota
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Summary

Citing D. Voiculescu, “Around 1982, I realized that the right way to look at certain operator algebra problems was by imitating some basic probability theory. More precisely, in noncommutative probability theory a new kind of independence can be defined by replacing tensor products with free products and this can help understand the von Neumann algebras of free groups. The subject has evolved into a kind of parallel to basic probability theory, which should be called free probability theory.

Thus, Voiculescu's first motivation to introduce free probability was the analysis of the von Neumann algebras of free groups. One of his central observations was that such groups can be equipped with tracial states (also called traces), which resemble expectations in classical probability, whereas the property of freeness, once properly stated, can be seen as a notion similar to independence in classical probability. This led him to the statement

free probability theory=noncommutative probability theory+ free independence.

These two components are the basis for a probability theory for noncommutative variables where many concepts taken from probability theory such as the notions of laws, convergence in law, independence, central limit theorem, Brownian motion, entropy and more can be naturally defined. For instance, the law of one self-adjoint variable is simply given by the traces of its powers (which generalizes the definition through moments of compactly supported probability measures on the real line), and the joint law of several self-adjoint noncommutative variables is defined by the collection of traces of words in these variables.

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Publisher: Cambridge University Press
Print publication year: 2009

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  • Free probability
  • Greg W. Anderson, University of Minnesota, Alice Guionnet, Ecole Normale Supérieure, Lyon, Ofer Zeitouni, Weizmann Institute/University of Minnesota
  • Book: An Introduction to Random Matrices
  • Online publication: 03 February 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511801334.006
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  • Free probability
  • Greg W. Anderson, University of Minnesota, Alice Guionnet, Ecole Normale Supérieure, Lyon, Ofer Zeitouni, Weizmann Institute/University of Minnesota
  • Book: An Introduction to Random Matrices
  • Online publication: 03 February 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511801334.006
Available formats
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Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Free probability
  • Greg W. Anderson, University of Minnesota, Alice Guionnet, Ecole Normale Supérieure, Lyon, Ofer Zeitouni, Weizmann Institute/University of Minnesota
  • Book: An Introduction to Random Matrices
  • Online publication: 03 February 2011
  • Chapter DOI: https://doi.org/10.1017/CBO9780511801334.006
Available formats
×