Fock representation of free convolution powers

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