Free convolution operators and free Hall transform

We define an extension of the polynomial calculus on a W^?$W^{?}$-probability space by introducing an algebra C{_Xi:i∈I}$\mathbb{C}\{X_i:i\in I\}$ 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)${\mathit{GL}}_N(\mathbb{C})$ to the ?-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U(N)$U(N)$ to the free Hall transform.