Poisson Transforms on SomeC*-Algebras Generated by Isometries

A noncommutative Poisson transform associated to a certain class of sequences of operators on Hilbert spaces, with property (P), is defined on some universalC*-algebras (resp. nonselfadjoint algebras) generated by isometries. Its properties are described and used to study these universal algebras and their representations. As consequences, we obtain a functional calculus, isometric (resp. unitary) dilations, and commutant lifting theorem for the class of sequences of operators with property (P). Our “geometrical” approach leads also to new and elementary proofs as well as extensions of some classical results.