Rectangular random matrices, related free entropy and free Fisher's information

We prove that independent rectangular random matrices, when embedded in a space of larger square matrices, are asymptotically free with amalgamation over a commutative finite dimensional subalgebra D (under an hypothesis of unitary invariance). Then we consider elements of a finite von Neumann algebra containing D, which have kernel and range projection in D. We associate them a free entropy with the microstates approach, and a free Fisher’s information with the conjugate variables approach. Both give rise to optimization problems whose solutions involve freeness with amalgamation over D. It could be a first proposition for the study of operators between different Hilbert spaces with the tools of free probability. As an application, we prove a result of freeness with amalgamation between the two parts of the polar decomposition of R-diagonal elements with non trivial kernel.