Operator-valued version of conditionally free product

Abstract

Abstract. We present an operator-valued version of the conditionally free product of states and measures, which in the scalar case was studied by Bożejko, Leinert and Speicher. The related combinatorics and limit theorems are provided.1. Introduction. The concept of free probability has been developed since the pioneering work of Voiculescu [V]. In this theory a probability space is a unital complex * -algebra A, elements of which are viewed as random variables, endowed with a state φ which plays the role of the expectation. Having a family (A i , φ i ) i∈I of such probability spaces we can define another one, (A, φ), where A is the unital free product * i∈I A i and φ is a state defined by the property that φ(a 1 . . . a m ) = 0 wheneverIn this situation one says that the family {A i } i∈I of subalgebras is free in the probability space (A, φ). This notion leads naturally to that of free convolution of probability measures on the real line. Speicher [S2, S3] has provided a combinatorial description of this theory by using the lattice of noncrossing partitions.Later on Bożejko, Leinert and Speicher [BS, BLS] investigated probability spaces A endowed with a pair (φ, ψ) of states. Having a family (A i , φ i , ψ i ) i∈I of such spaces they were able to construct a probability space (A, φ, ψ) (A, φ, ψ)). This led them to the notion of free convolution of pairs of probability measures on R.Here we are going to extend this theory to the situation when φ is allowed to be an operator-valued state, i.e. when φ is of the form φ(a) = P 0 π(a)| H 0 , where H 0 is a fixed Hilbert space, π is a * -representation of A in a Hilbert space H ⊃ H 0 and P 0 is the orthonormal projection of H onto H 0 , so that

The von Neumann algebras generated by t-gaussians

“…F is the conditional Fock space and Ω the vacuum state, we refer to [12] Section 6 for more details about this construction. Actually this is also a particular case of the construction of completely positive maps on full free products by Boca in [2,3] when the amalgamation is over the complex field.…”

“…-The first part of this lemma is just a restatement of the GNS construction (or Stingspring dilation as we are in the scalar case) of [12] or [3]. In particular, it can be used to show that φ is indeed a state (that is Theorem 2.2 in [6]).…”

A combinatorial approach to monotonic independence over a C∗-algebra

“…I am also indebted to Wojtech Młotkowski for the reference [Młotkowski 2002], but especially to Hari Bercovici for his priceless help in conceiving and writing this material.…”

“…Since in the scalar-valued case the density of the limit distribution is the arcsine function [Lu 1997;Muraki 2000], the limit in Theorem 5.3 can be regarded as an "operator-valued arcsine law". Section 6 introduces a notion of conditionally free product of conditional expectations extending the definition and positivity results from [Młotkowski 2002] and shows the connection to monotonic independence analogous to [Franz 2005, Proposition 3.1].…”

Irreducible representations of the free product of groups

“…One family of such constructions is related to the commutative convolution algebra of radial functions on the free group * i∈I Z, see [FP1,FP2,T2,K,KS1,MZ,PS], or on the free product of cyclic groups of the same order, see [IP, Wy]. On the other hand there are developed methods to produce a representation of G = * i∈I G i (or, more generally, of a unital free product A = * i∈I A i of * -algebras A i ) from those of G i 's (or A i 's), see [B1,Av,Vo,VDN,BS,BLS,M2,M3,M4,M5].…”