Point evaluation and Hardy space on a homogeneous tree

We consider transfer functions of time–invariant systems as defined by Basseville, Benveniste, Nikoukhah and Willsky when the discrete time is replaced by the nodes of an homogeneous tree. The complex numbers are now replaced by a C*-algebra built from the structure of the tree. We define a point evaluation with values in this C*-algebra and a corresponding ``Hardy space’’ in which a Cauchy’s formula holds. This point evaluation is used to define in this context the counterpart of classical notions such as Blaschke factors. There are deep analogies with the non stationary setting as developed by the first author, Dewilde and Dym.