Fock representation of free convolution powers

TL;DR

This paper introduces a Fock space construction that provides a new operator realization of free convolution powers, connecting free, Boolean, and conditionally free independence in a unified framework.

Contribution

It develops a novel Fock space approach to realize free convolution powers and relates different types of independence through a unified algebraic construction.

Findings

Provides a new operator realization of free convolution powers

Establishes connections between free, Boolean, and conditionally free independence

Computes several von Neumann algebras arising from the construction

Abstract

Let $B$ be a star-algebra with a state $\phi$, and $t > 0$. Through a Fock space construction, we define two states $\Phi_t$ and $\Psi_t$ on the tensor algebra $T(B, \phi)$ such that under the natural map $(B, \phi) \rightarrow (T(B, \phi), \Phi_t, \Psi_t)$, free independence of arguments leads to free independence, while Boolean independence of centered arguments leads to conditionally free independence. The construction gives a new operator realization of the $(1+t)$'th free convolution power of any joint (star) distribution. We also compute several von Neumann algebras which arise.