Noncommutative joint dilations and free product operator algebras

Abstract

Let An (n = 2, 3, . . . , or n = ∞) be the noncommutative disc algebra, and On (resp. Tn) be the Cuntz (resp. Toeplitz) algebra on n generators. Minimal joint isometric dilations for families of contractive sequences of operators on a Hilbert space are obtained and used to extend the von Neumann inequality and the commutant lifting theorem to our noncommutative setting. We show that the universal algebra generated by k contractive sequences of operators and the identity is the amalgamated free product operator algebra ∗CAni for some positive integers n1, n2, . . . , nk ≥ 1, and characterize the completely bounded representations of ∗CAni . It is also shown that ∗CAni is completely isometrically imbedded in the “biggest” free product C∗-algebra ∗CTni (resp. ∗COni), and that all these algebras are completely isometrically isomorphic to some universal free operator algebras, providing in this way some factorization theorems. We show that the free product disc algebra ∗CAni is not amenable and the set of all its characters is homeomorphic to (C1)1 × · · · × (Ck)1. An extension of the Naimark dilation theorem to free semigroups is considered. This is used to construct a large class of positive definite operator-valued kernels on the unital free semigroup on n generators and to study the class Cρ (ρ > 0) of ρ-contractive sequences of operators. The dilation theorems are also used to extend the operatorial trigonometric moment problem to the free product C∗-algebras ∗CTni and ∗COni .