Conditionally Bi-Free Independence with Amalgamation

Abstract

In this paper, we introduce the notion of conditionally bi-free independence in an amalgamated setting. We define operator-valued conditionally bi-multiplicative pairs of functions and construct operatorvalued conditionally bi-free moment and cumulant functions. It is demonstrated that conditionally bifree independence with amalgamation is equivalent to the vanishing of mixed operator-valued bi-free and conditionally bi-free cumulants. Furthermore, an operator-valued conditionally bi-free partial R-transform is constructed and various operator-valued conditionally bi-free limit theorems are studied. IntroductionThe notion of conditionally free (c-free for short) independence was introduced in [3] as a generalization of the notion of free independence to two-state systems. In our previous paper [6] we introduced the notion of conditionally bi-free (c-bi-free for short) independence in order to study the non-commutative left and right actions of algebras on a reduced c-free product simultaneously. Thus conditional bi-freeness is an extension of the notion of bi-free independence [14] to two-state systems. Moreover [6] introduced c-(ℓ, r)-cumulants and demonstrated that a family of pairs of algebras in a two-state non-commutative probability space is conditionally bi-free if and only if mixed (ℓ, r)-and c-(ℓ, r)-cumulants vanish.In [13] Voiculescu generalized his own notion of free independence by replacing the scalars with an arbitrary algebra thereby obtaining the notion of free independence with amalgamation (see also [12] for the combinatorial aspects). For c-free independence, the generalization to an amalgamated setting over a pair of algebras was done by Popa in [9] (see also [8]). On the other hand, the framework for generalizing bi-free independence to an amalgamated setting was suggested by Voiculescu in [14, Section 8] and the theory was fully developed in [4].The main goal of this paper is to extend the notion of c-bi-free independence to an amalgamated setting over a pair of algebras. Furthermore, we demonstrate that the combinatorics of conditionally bi-free probability and bi-free probability with amalgamation, which are governed by the lattice of bi-non-crossing partitions, are specific instances of more general combinatorial structures.Including this introduction this paper contains nine sections which are structured as follows. Section 2 briefly reviews some of the background material pertaining to conditionally bi-free probability and bi-free probability with amalgamation from [4][5][6]. In particular, the notions bi-non-crossing partitions and diagrams, their lateral refinements and cappings, interior and exterior blocks, B-B-non-commutative probability spaces, operator-valued bi-multiplicative functions, and the operator-valued bi-free moment and cumulant functions are recalled.Section 3 introduces the structures studied within conditionally bi-free independence with amalgamation. We define the notion of a B-B-non-commutative probability space with a pair of (B, D)-valued expectations (A, E, F, ...

Bi-Boolean Independence for Pairs of Algebras

“…. , m + n}, we obtain On the other hand, expanding M (Z ℓ ,Zr ) (b, d) using the fact it converges absolutely produces As the cbi-free, operator-valued c-bi-free, and bi-Boolean analogues of these results are available in [11,Section 4], [12,Section 7], and Section 4 above respectively, it can be shown without any difficulty that similar results also hold in the framework of the current section.…”

“…Furthermore, like any other non-commutative probability theory, whenever a notion of independence (scalar-valued or operator-valued) is defined, a number of limit theorems can be obtained for various purposes. Since we do not have any immediate applications of the operator-valued bi-Boolean limit theorems available at the moment, we will simply mention that the bi-Boolean analogues of the operator-valued bi-free/c-bi-free limit theorems in [12,Section 9] can be easily proved by the same techniques.…”

Free-Boolean independence with amalgamation

“…Note that Z * bf is of the same type as Z f b following Proposition-Defintion 6.9 (4), (5). It is easy to check that the following relation holds: Ψ(Z bf ) * = Ψ * (Z * bf ).…”

“…Hence E(Z f bf A) = 0. Similary, BZ f bf can be written as the summation of elemtns of the types (4) and (5). Hence E(BZ f bf ) = 0.…”

“…Recently, their corresponding independence relations for pairs of random variables, analog of Voiculescu's bi-free theory, were introduced and studied [4,3,6]. Furthermore, the conditionally bi-free independence with amalgamation is studied in [5].…”