Abstract

Let A_n (n = 2,3,…, or n = ∞) be the noncommutative disc algebra, and O_n (resp. T_n) 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 *_CA_ni for some positive integers n₁,n₂,…,n_k ≥ 1, and characterize the completely bounded representations of *_CA_ni. It is also shown that *_CA_ni is completely isometrically imbedded in the “biggest” free product C^*-algebra  *_CT_ni (resp. *_CO_ni), 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  *_CA_ni is not amenable and the set of all its characters is homeomorphic to (C^n₁)₁ ×⋯ ×(C^n_k)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 C_ρ (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 *_CT_ni and *_CO_ni.

Authors