An Operad of Non-commutative Independences Defined by Trees

Abstract

We study notions of N -ary non-commutative independence, which generalize free, Boolean, and monotone independence. For every rooted subtree T of the N -regular tree, we define the T -free product of N noncommutative probability spaces and the T -free additive convolution of N non-commutative laws.These N -ary convolution operations form a topological symmetric operad which includes the free, Boolean, monotone, and anti-monotone convolutions, as well as the orthogonal and subordination convolutions. Using the operadic framework, the proof of convolution identities such as µ ⊞ ν = µ (ν i µ) can be reduced to combinatorial manipulations of trees. In particular, we obtain a decomposition of the T -free convolution into iterated Boolean and orthogonal convolutions, which generalizes work of Lenczewski.We also develop a theory of T -free independence that closely parallels the free, Boolean, and monotone cases, provided that the root vertex has more than one neighbor. This includes combinatorial moment formulas, cumulants, a central limit theorem, and classification of infinitely divisible distributions (in the case of bounded support). 1/2 B defines a semi-norm on H. We also have hb ≤ h b for h ∈ H and b ∈ B.Definition 2.2. If H is a Banach space with respect to this norm, then we say that H is a right Hilbert B-module. In general, if H has a B-valued semi-inner product, then the completion of H/{h : h = 0} is a right Hilbert B-module with the right B-action and the B-valued inner product induced in the natural way from those of H. We refer to this module as the completed quotient of H with respect to •, • . Definition 2.3. Let H 1 and H 2 be Hilbert B-modules, we say that a linear map T : H 1 → H 2 is right B-modular if T (hb) = (T h)b for h ∈ H 1 and b ∈ B. We say that T is adjointable if there exists a map T * : H 2 → H 1 such that

Berry-Esseen bounds for the multivariate $\mathcal{B}$-free CLT and operator-valued matrices

“…Our approach produces the first quantitative bounds on the Lévy distance for the multivariate Bfree CLT without requiring any regularity conditions on the analytic distributions of the B-valued circular family. Note that quantitative results on the operator-valued Cauchy transforms for the CLT in the setting of T-free independence were obtained by Jekel and Liu [25]. The same rate of convergence as in our theorem can be obtained as a non-trivial consequence of their Theorem 8.10.…”

“…While their bound would improve on the power of ℑ(b) −1 , it yields on the other hand estimates in terms of the operator norm instead of the moments; cf. [25,Remark 8.15]. Our extension of the Lindeberg method to the operator-valued setting allows to obtain bounds depending only on the moments for linear matrix pencils.…”

“…The same is true for operator-valued free probability, where the fundamental notion of free independence is generalized to free independence with amalgamation as a kind of conditional version of the former. Its development naturally lead to operator-valued free analogues of key and fundamental limiting theorems such as the operator-valued free Central Limit Theorem (CLT) due to Voiculescu [51,46,11,25] and results about the asymptotic behaviour of distributions of matrices with operator-valued entries [50,42,43,41,30,6]. In this paper, we give quantitative versions of such limit theorems by providing bounds on the level of the operator-valued Cauchy transform and the Lévy distance.…”

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

“…Here is how the preceding proposition applies to some examples discussed in Section 7. We mention that a rather general result of this kind, going in a framework of cumulant constructions related to trees, appears as Lemma 7.6 of [16].…”

Boolean Cumulants and Subordination in Free Probability

“…For another interpretation of the subordination function as generating function of certain "taboo" probabilities in the context of random walks see [31,Proposition 4]. We also present a sketch of an alternative proof which uses recent characterization of freeness in terms of Boolean cumulants given in [15,17]. We then use this lemma to give a precise formula for the conditional expectation of certain resolvents.…”