On operator-valued bi-free distributions

Abstract

In this paper, operator-valued bi-free distributions are investigated. Given a subalgebra D of a unital algebra B, it is established that a two-faced family Z is bi-free from (B, B op ) over D if and only if certain conditions relating the B-valued and D-valued bi-free cumulants of Z are satisfied. Using this, we verify that a two-faced family of matrices is R-cyclic if and only if they are bi-free from the scalar matrices over the scalar diagonal matrices. Furthermore, the operator-valued bi-free partial R-, S-, and T -transforms are constructed. New proofs of results from free probability are developed in order to facilitate many of these bi-free results.for all T ∈ L(M); that is, apply T to 1 M and take the expectation of M onto N. Further define *homomorphisms L : M → L(M) and R : M op → L(M) by L(X)(A) = XA and R(X)(A) = AX for all X, A ∈ M. For this discussion we will call L(X) a left operator and R(X) a right operator. If M = M 1 * N M 2 , the amalgamated free product of von Neumann algebras M 1 and M 2 over N, then the Date: July 9, 2018. 2010 Mathematics Subject Classification. 46L54, 46L53. Key words and phrases. bi-free probability, operator-valued distributions, amalgamating over a subalgebra, R-cyclic matrices, R-transform, S-transform. PAUL SKOUFRANIS Given χ : {1, . . . , n} → {ℓ, r} withdefine the permutation s χ on {1, . . . , n} by s χ (q) = k q . The only differences between the combinatorial aspects of free and bi-free probability arise from dealing with s χ . Using s χ , define the total ordering ≺ χ on {1, . . . , n} by k 1 ≺ χ k 2 if and only if s −1 χ (k 1 ) < s −1 χ (k 2 ). Instead of reading {1, . . . , n} in the traditional order, ≺ χ corresponds to reading χ −1 ({ℓ}) in increasing order followed by reading χ −1 ({r}) in decreasing order.A subset V ⊆ {1, . . . , n} is said to be a χ-interval if V is an interval with respect to the ordering ≺ χ . In addition, min ≺χ (V ) and max ≺χ (V ) denote the minimal and maximal elements of V with respect to the ordering ≺ χ .Definition 2.1. A partition π ∈ P(n) is said to be bi-non-crossing with respect to χ if the partition s −1 χ · π (the partition formed by applying s −1 χ to each entry of each block of π) is non-crossing. Equivalently π is bi-non-crossing if whenever there are blocks U, V ∈ π with u 1

Bi-Boolean Independence for Pairs of Algebras

“…. , R d Z r n times ).By similar arguments as in[23, Remark 5.2], all of the series above converge absolutely for b, d near 0. It is known (see, e.g.,[20, Section 4]) that M ℓ Z ℓ (b) and M r Zr (d) are invertible and that η ℓ Z ℓ (b) = 1 − M ℓ Z ℓ (b) −1 and η r Zr (d) = 1 − M r Zr (d) −1 for b, d ∈ Bsufficiently small.…”

Analytic subordination for bi-free convolution

“…Consequently, a C * -, or a W * -B-B noncommutative probability space requires, beyond the data (M, E, B) described above, a way to view B and B op simultaneously as subalgebras of M , that is, a linear homomorphism ε : B ⊗ B op → M . This homomorphism satisfies certain conditions for which we refer to [24,Definition 2.5] (see also [15]). For our study, it is important to note that we must add an ℓ or an r to each analytic transform defined on B, corresponding to whether it is viewed as being defined on B or B op .…”

“…In this context, our interest is mainly in bi-freeness with amalgamation over finite dimensional algebras. The construction providing an M n (C)-M n (C)-noncommutative probability space from a classical noncommutative probability space (A, ϕ) is the following (see [23,Section 6] or [24,Section 4]). First, one defines the left and right actions…”

Faithfulness of bifree product states

“…It should be mentioned that the failure in general of the faithfulness of the bi-free product state has been pointed out in [4] and this failure has been the cause of the introduction of weaker versions of faithfulness in the bi-free context [4,5].…”