1,008 research outputs found
Quantum Exchangeable Sequences of Algebras
We extend the notion of quantum exchangeability, introduced by K\"ostler 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.Comment: Added comments and reference [8], final version to appear in Indiana
Univ. Math. Journa
Quantum invariant families of matrices in free probability
We consider (self-adjoint) families of infinite matrices of noncommutative
random variables such that the joint distribution of their entries is invariant
under conjugation by a free quantum group. For the free orthogonal and
hyperoctahedral groups, we obtain complete characterizations of the invariant
families in terms of an operator-valued -cyclicity condition. This is a
surprising contrast with the Aldous-Hoover characterization of jointly
exchangeable arrays.Comment: 33 page
Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices
We consider the limiting distribution of and (and more
general expressions), where and are matrices with
entries in a unital C-algebra which have limiting -valued distributions as , and is a Haar
distributed quantum unitary random matrix with entries independent from
. Under a boundedness assumption, we show that and
are asymptotically free with amalgamation over . Moreover,
this also holds in the stronger infinitesimal sense of Belinschi-Shlyakhtenko.
We provide an example which demonstrates that this example may fail for
classical Haar unitary random matrices when the algebra is
infinite-dimensional.Comment: Added reference [13], and replaced Lemma 3.7 by a stronger result
from that paper. Minor change to the statement of Theorem 4.6. 25 pages, 3
figure
Stochastic aspects of easy quantum groups
We consider several orthogonal quantum groups satisfying the easiness
assumption axiomatized in our previous paper. For each of them we discuss the
computation of the asymptotic law of Tr(u^k) with respect to the Haar measure,
u being the fundamental representation. For the classical groups O_n, S_n we
recover in this way some well-known results of Diaconis and Shahshahani.Comment: 28 page
De Finetti theorems for easy quantum groups
We study sequences of noncommutative random variables which are invariant
under "quantum transformations" coming from an orthogonal quantum group
satisfying the "easiness" condition axiomatized in our previous paper. For 10
easy quantum groups, we obtain de Finetti type theorems characterizing the
joint distribution of any infinite quantum invariant sequence. In particular,
we give a new and unified proof of the classical results of de Finetti and
Freedman for the easy groups S_n, O_n, which is based on the combinatorial
theory of cumulants. We also recover the free de Finetti theorem of K\"ostler
and Speicher, and the characterization of operator-valued free semicircular
families due to Curran. We consider also finite sequences, and prove an
approximation result in the spirit of Diaconis and Freedman.Comment: Published in at http://dx.doi.org/10.1214/10-AOP619 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- β¦