Free convolution operators and free Hall transform

We define an extension of the polynomial calculus on a W⁎-probability space by introducing an algebra C{Xi:i∈I} which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal-Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N tends to ∞, of the ⁎-distribution of the Brownian motion on the linear group GLN(C) to the ⁎-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U(N) to the free Hall transform.