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De Finetti Theorems for the Unitary Dual Group
Symmetry, integrability and geometry: methods and applications, Tome 18 (2022) Cet article a éte moissonné depuis la source Math-Net.Ru

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We prove several de Finetti theorems for the unitary dual group, also called the Brown algebra. Firstly, we provide a finite de Finetti theorem characterizing $R$-diagonal elements with an identical distribution. This is surprising, since it applies to finite sequences in contrast to the de Finetti theorems for classical and quantum groups; also, it does not involve any known independence notion. Secondly, considering infinite sequences in $W^*$-probability spaces, our characterization boils down to operator-valued free centered circular elements, as in the case of the unitary quantum group $U_n^+$. Thirdly, the above de Finetti theorems build on dual group actions, the natural action when viewing the Brown algebra as a dual group. However, we may also equip the Brown algebra with a bialgebra action, which is closer to the quantum group setting in a way. But then, we obtain a no-go de Finetti theorem: invariance under the bialgebra action of the Brown algebra yields zero sequences, in $W^*$-probability spaces. On the other hand, if we drop the assumption of faithful states in $W^*$-probability spaces, we obtain a non-trivial half a de Finetti theorem similar to the case of the dual group action.
Keywords: de Finetti theorem, distributional invariance, exchangeable, Brown algebra, unitary dual group, free circular elements.
Mots-clés : $R$-diagonal elements 
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Isabelle Baraquin; Guillaume Cébron; Uwe Franz; Laura Maassen; Moritz Weber. De Finetti Theorems for the Unitary Dual Group. Symmetry, integrability and geometry: methods and applications, Tome 18 (2022). http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a66/

[1] Arnon-Friedman R., Renner R., Vidick T., “Non-signaling parallel repetition using de Finetti reductions”, IEEE Trans. Inform. Theory, 62 (2016), 1440–1457, arXiv: 1411.1582 | DOI | MR

[2] Banica T., Curran S., Speicher R., “De Finetti theorems for easy quantum groups”, Ann. Probab., 40 (2012), 401–435, arXiv: 0907.3314 | DOI | MR

[3] Brandão F.G.S.L., Harrow A.W., “Quantum de Finetti theorems under local measurements with applications”, Comm. Math. Phys., 353 (2017), 469–506, arXiv: 1210.6367 | DOI | MR

[4] Brown L.G., “Ext of certain free product $C^{\ast} $-algebras”, J. Operator Theory, 6 (1981), 135–141 | MR

[5] Cébron G., Ulrich M., “Haar states and Lévy processes on the unitary dual group”, J. Funct. Anal., 270 (2016), 2769–2811, arXiv: 1505.08083 | DOI | MR

[6] Christandl M., König R., Mitchison G., Renner R., “One-and-a-half quantum de Finetti theorems”, Comm. Math. Phys., 273 (2007), 473–498, arXiv: quant-ph/0602130 | DOI | MR

[7] Curran S., “Quantum rotatability”, Trans. Amer. Math. Soc., 362 (2010), 4831–4851, arXiv: 0901.1855 | DOI | MR

[8] Curran S., “A characterization of freeness by invariance under quantum spreading”, J. Reine Angew. Math., 659 (2011), 43–65, arXiv: 1002.4390 | DOI | MR

[9] de Finetti B., “La prévision: ses lois logiques, ses sources subjectives”, Ann. Inst. H. Poincaré, 7 (1937), 1–68 | MR

[10] Freedman D.A., “Invariants under mixing which generalize de Finetti's theorem”, Ann. Math. Statist., 33 (1962), 916–923 | DOI | MR

[11] Freslon A., Weber M., “On bi-free de Finetti theorems”, Ann. Math. Blaise Pascal, 23 (2016), 21–51 | DOI | MR

[12] Glockner P., von Waldenfels W., “The relations of the noncommutative coefficient algebra of the unitary group”, Quantum Probability and Applications, IV (Rome, 1987), Lecture Notes in Math., 1396, Springer, Berlin, 1989, 182–220 | DOI | MR

[13] Gohm R., Köstler C., “Noncommutative independence from the braid group $\mathbb B_\infty$”, Comm. Math. Phys., 289 (2009), 435–482, arXiv: 0806.3691 | DOI | MR

[14] Gohm R., Köstler C., “Noncommutative independence in the infinite braid and symmetric group”, Noncommutative Harmonic Analysis with Applications to Probability III, Banach Center Publ., 96, Polish Acad. Sci. Inst. Math., Warsaw, 2012, 193–206, arXiv: 1102.0813 | DOI | MR

[15] Hayase T., “De Finetti theorems for a Boolean analogue of easy quantum groups”, J. Math. Sci. Univ. Tokyo, 24 (2017), 355–398, arXiv: 1507.05563 | MR

[16] Hudson R.L., “Analogs of de Finetti's theorem and interpretative problems of quantum mechanics”, Found. Phys., 11 (1981), 805–808 | DOI | MR

[17] Kallenberg O., Probabilistic symmetries and invariance principles, Probability and its Applications (New York), Springer, New York, 2005 | DOI | MR

[18] Köstler C., “A noncommutative extended de Finetti theorem”, J. Funct. Anal., 258 (2010), 1073–1120, arXiv: 0806.3621 | DOI | MR

[19] Köstler C., Speicher R., “A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation”, Comm. Math. Phys., 291 (2009), 473–490, arXiv: 0807.0677 | DOI | MR

[20] Lancien C., Winter A., “Flexible constrained de Finetti reductions and applications”, J. Math. Phys., 58 (2017), 092203, 24 pp., arXiv: 1605.09013 | DOI | MR

[21] Liu W., “A noncommutative de Finetti theorem for boolean independence”, J. Funct. Anal., 269 (2015), 1950–1994, arXiv: 1403.1772 | DOI | MR

[22] Liu W., “Extended de Finetti theorems for boolean independence and monotone independence”, Trans. Amer. Math. Soc., 370 (2018), 1959–2003, arXiv: 1505.02215 | DOI | MR

[23] Liu W., “General de Finetti type theorems in noncommutative probability”, Comm. Math. Phys., 369 (2019), 837–866, arXiv: 1511.05651 | DOI | MR

[24] Maassen L., Representation categories of compact matrix quantum groups, Ph.D. Thesis, RWTH Aachen University, 2021 | DOI

[25] McClanahan K., “$C^*$-algebras generated by elements of a unitary matrix”, J. Funct. Anal., 107 (1992), 439–457 | DOI | MR

[26] Mingo J.A., Speicher R., Free probability and random matrices, Fields Institute Monographs, 35, Springer, New York, 2017 | DOI | MR

[27] Nica A., Shlyakhtenko D., Speicher R., “Operator-valued distributions. I Characterizations of freeness”, Int. Math. Res. Not., 2002 (2002), 1509–1538, arXiv: math.OA/0201001 | DOI | MR

[28] Nica A., Speicher R., Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, Cambridge, 2006 | DOI | MR

[29] Raum S., Weber M., “Easy quantum groups and quantum subgroups of a semi-direct product quantum group”, J. Noncommut. Geom., 9 (2015), 1261–1293, arXiv: 1311.7630 | DOI | MR

[30] Sołtan P.M., “Quantum families of maps and quantum semigroups on finite quantum spaces”, J. Geom. Phys., 59 (2009), 354–368, arXiv: math.OA/0610922 | DOI | MR

[31] Takesaki M., Theory of operator algebras, v. II, Encyclopaedia of Mathematical Sciences, 125, Springer-Verlag, Berlin, 2003 | DOI | MR

[32] Voiculescu D., “Dual algebraic structures on operator algebras related to free products”, J. Operator Theory, 17 (1987) | MR

[33] Wang S., “Free products of compact quantum groups”, Comm. Math. Phys., 167 (1995), 671–692 | DOI | MR