Quantum invariant families of matrices in free probability

Abstract

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 R-cyclicity condition. This is a surprising contrast with the Aldous-Hoover characterization of jointly exchangeable arrays.

The algebraic structure of quantum partial isometries

“…There is a certain similarity between the present work and the one in [7], waiting to be understood, and axiomatized. (3) Finally, the quantum groups S + N , O + N , U + N have been successfully used in connection with several free probability questions [17], [18], [20], [21]. The semigroup extension of these results is an open problem, that we would like to raise here.…”

Liberation theory for noncommutative homogeneous spaces

“…(2) A second question concerns the validity of the quantum isometry group formula G + (X + ) = G(X) + , in relation with the rigidity results in [11], [16]. (3) Yet another question regards the possible applications of the present formalism to free probability invariance questions, in the spirit of [13], [18]. (4) Finally, there are as well several interesting questions in relation with the axiomatization problem for the noncommutative algebraic manifolds [17].…”

A duality principle for noncommutative cubes and spheres

“…The whole idea ended up in producing a very active field of research. See [20], [32], [36], [37], [38], [46], [47], [48], [49].…”