FOCK REPRESENTATION OF FREE CONVOLUTION POWERS

Let B be a & lowast;-algebra with a state phi , and t > 0. Through a Fock space construction, we define two states Phi(t) and Psi t t on the tensor algebra T (B,phi) such that under the natural map (B,phi) -> (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.