⁎-freeness in finite tensor products

Abstract

Abstract. In this paper, we consider the following question and variants thereof: given D := a 1;i ⊗· · ·⊗a K;i : i ∈ I , a collection of elementary tensor non-commutative random variables in the tensor product of probability spaces (when is D * -free? (See Section 1.2 for a precise formulation of this problem.)Settling whether or not freeness occurs in tensor products is a recurring problem in operator algebras, and the following two examples provide a natural motivation for the above question: (A) If (a 1;i : i ∈ I) is a * -free family of Haar unitary variables and a k,i are arbitrary unitary variables for k ≥ 2, then the * -freeness persists at the level of the tensor product D. (B) A converse of (A) holds true if all variables a k;i are group-like elements (see Corollary 1.7 of Proposition 1.6). It is therefore natural to seek to understand the extent to which such simple characterizations hold true in more general cases. While our results fall short of a complete characterization, we make notable steps toward identifying necessary and sufficient conditions for the freeness of D. For example, we show that under evident assumptions, if more than one family (a k,i : i ∈ I) contains non-unitary variables, then the tensor family fails to be * -free (see Theorem 1.8 (1)).

Eigenvalues of random lifts and polynomials of random permutation matrices

“…Note that asymptotic freeness follows, among others, from [13]. In particular, when one restricts oneself to sum of generators i S i ⊗ S i , one obtains that the family S i ⊗ S i viewed as an operator on V ⊥ is a nearly optimal quantum expander in the sense of Hastings [26] and Pisier [38].…”

“…, U d ) are independent Haar unitaries on U N . For more details about this property (a variant of an absorption principle), we refer to [52,21,23].…”

Strong asymptotic freeness for independent uniform variables on compact groups associated to non-trivial representations

“…It is not a priori obvious why the claim ( 8) is easier to prove than claim (6). A reason is that the powers of (B ⋆ ) µ are much simpler to compute than the powers of A ⋆ .…”

“…As soon as compact matrix groups are involved, the joint asymptotic behavior of independent random variables can legitimately be expected to depend on which representation of the group is being considered, and for example, although the above results are conclusive examples of asymptotic freeness irrespective of the representation in the case of the symmetric group, we are not aware of any complete result in this direction for unitary or orthogonal groups, and the initial results deal rather with the very particular case of the fundamental representation. In a recent development by the second author together with Gaudreau Lamarre and Male addressed part of the problem in [6], in the case when a signature is fixed.…”