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General De Finetti Type Theorems in Noncommutative Probability

Communications in Mathematical Physics volume 369, pages 837–866 (2019)Cite this article

Abstract

We prove general de Finetti type theorems for classical and free independence. The de Finetti type theorems work for all non-easy quantum groups, which generalize a recent work of Banica, Curran and Speicher. We determine maximal distributional symmetries which means the corresponding de Finetti type theorem fails if a sequence of random variables satisfy more symmetry relations other than the maximal one. In addition, we define Boolean quantum semigroups in analogous to the easy quantum groups, by universal conditions on matrix coordinate generators and an orthogonal projection. Then, we show a general de Finetti type theorem for Boolean independence.

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Acknowledgement

The author would like to thank his thesis advisor, D.-V. Voiculescu, for his continued guidance and support while completing this project.While working on this paper, the author was supported in part by funds from NSF grant DMS-1301727.

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  1. Department of Mathematics, Indiana University, Bloomington, IN, 47401, USA

    Weihua Liu

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  1. Weihua Liu
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Correspondence to Weihua Liu.

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Communicated by Y. Kawahigashi

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Liu, W. General De Finetti Type Theorems in Noncommutative Probability. Commun. Math. Phys. 369, 837–866 (2019). https://doi.org/10.1007/s00220-019-03471-y

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  • DOI: https://doi.org/10.1007/s00220-019-03471-y