Discrete Approximation of the Free Fock Space

Abstract

We prove that the free Fock space $F(L^2 (\mathbb{R}^+;\mathbb{C}))$, which is very commonly used in Free Probability Theory, is the continuous free product of copies of the space $\mathbb{C}^2$. We describe an explicit embedding and approximation of this continuous free product structure by means of a discrete-time approximation: the free toy Fock space, a countable free product of copies of $\mathbb{C}^2$. We show that the basic creation, annihilation and gauge operators of the free Fock space are also limits of elementary operators on the free toy Fock space. When applying these constructions and results to the probabilistic interpretations of these spaces, we recover some discrete approximations of the semi-circular Brownian motion and of the free Poisson process. All these results are also extended to the higher multiplicity case, that is, $F(L^2(\mathbb{R}^+;\mathbb{C}^N))$ is the continuous free product of copies of the space $\mathbb{C}^{N+1}$.

2000 Mathematics Subject Classification: Primary 46L54. Secondary 46L09, 60F05