Let 1 be a free nonabelian group and let 0 be its boundary. Let ? h be one of the unitary representations of 1 introduced earlier by the authors in (1996, Duke Math. J. 82, 381 436). By its definition ? h acts on L 2 (0, d& h ) for a certain measure & h . This gives a boundary realization of ? h in a sense we make precise. We show that ? h does not have any other boundary realizations and simultaneously provide a new proof that ? h is irreducible. Academic Press

Abstract. We consider a countable discrete group F acting ergodically on a standard Borel space S with quasi-invariant measure p . Let n be a unitary representation of T on L2(S, dp, «#") "nicely" related with S. We prove that if T acts amenably on S then n is weakly contained in the regular representation.

Let (Formula presented.) be a non-commutative free group on finitely many generators. In a previous work two of the authors have constructed the class of multiplicative representations of (Formula presented.) and proved them irreducible as representation of (Formula presented.). In this paper we analyze multiplicative representations as representations of (Formula presented.) and we prove a criterium for irreducibility based on the growth of their matrix coefficients

Abstract. We consider a countable discrete group F acting ergodically on a standard Borel space S with quasi-invariant measure p . Let n be a unitary representation of T on L2(S, dp, «#") "nicely" related with S. We prove that if T acts amenably on S then n is weakly contained in the regular representation.

We present a powerful theorem for proving the irreducibility of tempered unitary representations of the free group.