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Operator models and analytic subordination for operator-valued free convolution powers

Ian Charlesworth, David Jekel

TL;DR

The paper develops a comprehensive operator-valued framework for free convolution powers μ^{⊞η} driven by a completely positive map η. It provides a general analytic realization: under suitable freeness and structural conditions, the η-convolution power of μ is realized by the law of $V^* X V$, unifying scalar (Nica–Speicher) and operator-valued (Shlyakhtenko) constructions, and it establishes an operator-valued subordination principle via a function $F$ with $G_ u = G_ ext{μ} igcirc F$ and a conditional expectation identity. A constructive approach yields an explicit model for V on ${ m B} igoplus ({ m B} frac{ m id}{ m B})$ using $oldsymbol ψ = η - { m id}$, and the proof proceeds by analyzing the functional equation for the $R$-transform and exploiting freeness to show $G_ u(z)$ satisfies the defining relation. The paper also clarifies the connection to the classical $n$-fold free convolution by exhibiting a precise transformation of ${ m B}$-${ m B}$-correspondences that demonstrates the two constructions yield the same distribution, thereby bridging the additive and η-powered viewpoints and enriching the toolbox for operator-valued free probability.

Abstract

We revisit the theory of operator-valued free convolution powers given by a completely positive map $η$. We first give a general result, with a new analytic proof, that the $η$-convolution power of the law of $X$ is realized by $V^*XV$ for any operator $V$ satisfying certain conditions, which unifies Nica and Speicher's construction in the scalar-valued setting and Shlyakhtenko's construction in the operator-valued setting. Second, we provide an analog, for the setting of $η$-valued convolution powers, of the analytic subordination for conditional expectations that holds for additive free convolution. Finally, we describe a Hilbert-space manipulation that explains the equivalence between the $n$-fold additive free convolution and the convolution power with respect to $η= n \operatorname{id}$.

Operator models and analytic subordination for operator-valued free convolution powers

TL;DR

The paper develops a comprehensive operator-valued framework for free convolution powers μ^{⊞η} driven by a completely positive map η. It provides a general analytic realization: under suitable freeness and structural conditions, the η-convolution power of μ is realized by the law of , unifying scalar (Nica–Speicher) and operator-valued (Shlyakhtenko) constructions, and it establishes an operator-valued subordination principle via a function with and a conditional expectation identity. A constructive approach yields an explicit model for V on using , and the proof proceeds by analyzing the functional equation for the -transform and exploiting freeness to show satisfies the defining relation. The paper also clarifies the connection to the classical -fold free convolution by exhibiting a precise transformation of --correspondences that demonstrates the two constructions yield the same distribution, thereby bridging the additive and η-powered viewpoints and enriching the toolbox for operator-valued free probability.

Abstract

We revisit the theory of operator-valued free convolution powers given by a completely positive map . We first give a general result, with a new analytic proof, that the -convolution power of the law of is realized by for any operator satisfying certain conditions, which unifies Nica and Speicher's construction in the scalar-valued setting and Shlyakhtenko's construction in the operator-valued setting. Second, we provide an analog, for the setting of -valued convolution powers, of the analytic subordination for conditional expectations that holds for additive free convolution. Finally, we describe a Hilbert-space manipulation that explains the equivalence between the -fold additive free convolution and the convolution power with respect to .
Paper Structure (8 sections, 10 theorems, 74 equations)

This paper contains 8 sections, 10 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. ℬ-valued non-commutative probability spaces
  4. ℬ-valued laws and analytic transforms
  5. ℬ-valued free independence and free products
  6. Realization of operator-valued convolution powers
  7. Subordination and conditional expectations
  8. On convolution powers and sums

Key Result

Lemma 2.6

Let ${\mathcal{A}}$ and ${\mathcal{B}}$ be $\mathrm{C}^*$-algebras and $\psi: {\mathcal{A}} \to {\mathcal{B}}$ be completely positive. On the algebraic tensor product ${\mathcal{A}} \otimes_{\mathop{\mathrm{alg}}\nolimits} {\mathcal{B}}$, define a ${\mathcal{B}}$-valued sesquilinear map by Then $\langle\cdot,\cdot\rangle_\psi$ is a ${\mathcal{B}}$-valued pre-inner product. The associated separati

Theorems & Definitions (27)

  • Definition 2.1: ${\mathcal{B}}$-valued probability space
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6: Paschke1973, see also Lance1995
  • Lemma 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10: ${\mathcal{B}}$-valued laws, as in Voiculescu1995PV2013AW2016
  • ...and 17 more