Quantum rotatability

Author:
Stephen Curran

Journal:
Trans. Amer. Math. Soc. **362** (2010), 4831-4851

MSC (2000):
Primary 46L54; Secondary 46L65, 60G09

DOI:
https://doi.org/10.1090/S0002-9947-10-05119-6

Published electronically:
April 27, 2010

MathSciNet review:
2645052

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Recently, Köstler and Speicher showed that de Finetti's theorem on exchangeable sequences has a free analogue if one replaces exchangeability by the stronger condition of invariance of the joint distribution under quantum permutations. In this paper we study sequences of noncommutative random variables whose joint distribution is invariant under quantum orthogonal transformations. We prove a free analogue of Freedman's characterization of conditionally independent Gaussian families; namely, the joint distribution of an infinite sequence of self-adjoint random variables is invariant under quantum orthogonal transformations if and only if the variables form an operator-valued free centered semicircular family with common variance. Similarly, we show that the joint distribution of an infinite sequence of random variables is invariant under quantum unitary transformations if and only if the variables form an operator-valued free centered circular family with common variance.

We provide an example to show that, as in the classical case, these results fail for finite sequences. We then give an approximation for how far the distribution of a finite quantum orthogonally invariant sequence is from the distribution of an operator-valued free centered semicircular family with common variance.

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Additional Information

**Stephen Curran**

Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

Email:
curransr@math.berkeley.edu

DOI:
https://doi.org/10.1090/S0002-9947-10-05119-6

Keywords:
Free probability,
quantum rotatability,
quantum invariance,
semicircle law

Received by editor(s):
February 11, 2009

Published electronically:
April 27, 2010

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.