Quantum Exchangeable Sequences of Algebras
Abstract
We extend the notion of quantum exchangeability, introduced by Köstler and Speicher in arXiv:0807.0677, to sequences (\rho_1,\rho_2,...c) of homomorphisms from an algebra C into a noncommutative probability space (A,\phi), and prove a free de Finetti theorem: an infinite quantum exchangeable sequence (\rho_1,\rho_2,...c) is freely independent and identically distributed with respect to a conditional expectation. As a corollary we obtain a free analogue of the Hewitt Savage zero-one law. As in the classical case, the theorem fails for finite sequences. We give a characterization of finite quantum exchangeable sequences, which can be viewed as a noncommutative analogue of sampling without replacement. We then give an approximation to how far a finite quantum exchangeable sequence is from being freely independent with amalgamation.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2008
- DOI:
- 10.48550/arXiv.0812.3428
- arXiv:
- arXiv:0812.3428
- Bibcode:
- 2008arXiv0812.3428C
- Keywords:
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- Mathematics - Operator Algebras;
- Mathematics - Quantum Algebra;
- 46L54;
- 46L65;
- 60G09
- E-Print:
- Added comments and reference [8], final version to appear in Indiana Univ. Math. Journal