Fock representation of free convolution powers

Abstract

Let $B$ be a star-algebra with a state $\varphi$, and $t>0$. Through a Fock space construction, we define two states ${\Phi }_t$ and ${\Psi }_t$ on the tensor algebra $T(B,\varphi )$ such that under the natural map $(B,\varphi )\to (T(B,\varphi ),{\Phi }_t,{\Psi }_t)$, 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.

v2: minor revision following comments by the referee

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