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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 (Cn1)1 ×⋯ ×(Cnk)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.
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