Induced Stinespring factorization and the Wittstock support theorem

TL;DR

This paper explores the structure of completely bounded maps on $C^*$-algebras, introducing a new support theorem for extremal Wittstock decompositions, with implications for transfer function realizations.

Contribution

It establishes the uniqueness of the support of extremal Wittstock decompositions and connects colligation formulas with transfer function realizations in this context.

Findings

Support of extremal Wittstock decomposition is unique.

Provides colligation formulae for maps in the Agler class.

Connects transfer function realizations with Wittstock decompositions.

Abstract

Given a pair of self-adjoint-preserving completely bounded maps on the same $C^*$-algebra, say that $\varphi \leq \psi$ if the kernel of $\varphi$ is a subset of the kernel of $\psi$ and $\psi \circ \varphi^{-1}$ is completely positive. The \emph{Agler class} of a map $\varphi$ is the class of $\psi \geq \varphi.$ Such maps admit colligation formulae, and, in Lyapunov type situations, transfer function type realizations on the Stinespring coefficients of their Wittstock decompositions. As an application, we prove that the support of an extremal Wittstock decomposition is unique.