Invariant structure preserving functions and an Oka-Weil Kaplansky density type theorem

Abstract

We develop the theory of invariant structure preserving and free functions on a general structured topological space. We show that an invariant structure preserving function is pointwise approximiable by the appropriate analog of polynomials in the strong topology and therefore a free function. Moreover, if a domain of operators on a Hilbert space is polynomially convex, the set of free functions satisfies a Oka-Weil Kaplansky density type theorem-contractive functions can be approximated by contractive polynomials.

Induced Stinespring factorization and the Wittstock support theorem

“…Some work shows that if Z is a matrix with z i on the diagonal and Λ with λ i on the diagonal, the corresponding Choi matrix is exactly the matrix arising in the classical Nevanlinna-Pick interpolation theorem. The formulation of the problem in terms of complete positivity of some induced map is a powerful idea which has led to broad generalizations in noncommuting variables, especially in terms of the work of Ball-Groenwald-Malakorn [6,5] and subsequent developments [8,7,9,11].…”

“…We say ϕ is Archimedian if its range contains a strictly positive element. The Agler order captures the various Agler models used for Nevanlinna-Pick interpolation in the Schur-Agler, Herglotz-Agler, Pick-Agler classes and so on, codifying the Lyapunov formulation taken in [6,5,8,7,9,11]. For example, taking ϕ(H) = H − X * HX, ψ(H) = Y * H + HY, we have that ϕ ≤ ψ corresponds to there being an analytic function from the disk to the right half plane (a Herglotz function) taking X to Y.…”

“…The compatibility of representations is important in infinite dimensional noncommutative interpolation problems, where there is some work to show the abstract technique here gives a bona fide solution (see, for example, the last section of [11]).…”

“…where U is a partial isometry (that is, ( This setup gives the 2-variable commutative Nevanlinna-Pick interpolation theorem. Similarly, as in [11], our method works on more general semi-algebraic sets in the general noncommutative case. Notice that ψ is scalar-valued, and so positivity of ψ • ϕ −1 implies complete positivity in this case.…”

“…Then (5equationsY = A + Bπ(X)Γ Γ = C + Dπ(X)Γ.Eliminating Γ gives the (Schur-Agler) transfer function realization(5.3) Y = A + Bπ(X)(1 − Dπ(X)) −1 C,which points to the existence of an interpolating function in terms of complete positivity. (c.f [3,4,11]…”