We characterize the minimal isometric dilation of a non-commutative contractive sequence of operators as a universal object for certain diagrams of completely positive maps. A non-spatial construction of the minimal isometric dilation is also given, using Hilbert modules over C*-algebras.

It is shown that the non-commutative disc algebras [Ascr ]n (n[ges ]2) are the universal algebras generated by contractive sequences of operators and the identity, and C*(S1, …, Sn) (n[ges ]2), the extension through compact operators of the Cuntz algebra [Oscr ]n, is the universal C*-algebra generated by a contractive sequence of isometries. It is also shown that the algebras [Ascr ]n and C*(S1, …, Sn) are completely isometrically isomorphic to some free operator algebras considered by D. Blecher. In particular, the universal operator algebra of a row (respectively column) contraction is identified with a subalgebra of C*(S1, …, Sn). The internal characterization of the matrix norm on a universal algebra leads to some factorization theorems.