An operator algebraic proof of Agler's factorization theorem

Factorizations of Schur functions

The Schur class, denoted by $\mathcal{S}(\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$, that is \[ \mathcal{S}(\mathbb{D}) = \{φ\in H^\infty(\mathbb{D}): \|φ\|_{\infty} := \sup_{z \in \mathbb{D}} |φ(z)| \leq 1\}. \] The elements of $\mathcal{S}(\mathbb{D})$ are called Schur functions. A classical result going back to I. Schur states: A function $φ: \mathbb{D} \rightarrow \mathbb{C}$ is in $\mathcal{S}(\mathbb{D})$ if and only if there exist a Hilbert space $\mathcal{H}$ and an isometry (know…

Induced Stinespring factorization and the Wittstock support theorem

Given a pair of self-adjoint-preserving completely bounded maps on the same $C^*$-algebra, say that $φ\leq ψ$ if the kernel of $φ$ is a subset of the kernel of $ψ$ and $ψ\circ φ^{-1}$ is completely positive. The \emph{Agler class} of a map $φ$ is the class of $ψ\geq φ.$ 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.

Realizations via Preorderings with Application to the Schur Class

We extend Agler's notion of a function algebra defined in terms of test functions to include products, in analogy with the practice in real algebraic geometry, and hence the term preordering in the title. This is done over abstract sets and no additional property, such as analyticity, is assumed. Realization theorems give several equivalent ways of characterizing the unit ball (referred to as the Schur-Agler class) of the function algebras. These typically include, in Agler's terminology, a model (here called an Agler decomposition), a transfer function representation, and an analogue of the v…

Operator Algebras of Functions

We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are …

C*-代数分解同调与循环表示的实现

C*-Algebraic Factorization Homology and Realization of Cyclic Representations

该研究证明C*-范畴max-张量积的余连续性，构建C*-框架下的分解同调框架，细化已有结果，实现量子哈密顿约化并提出强化的GNS型重构定理。

正定核的分解：$C^{*}$-代数与算子值语境与标量值核的对应关系

Factorization of positive definite kernels. Correspondences: $C^{*}$-algebraic and operator valued context vs scalar valued kernels

本文引入一类广义正定核，通过标量值核和RKHS方法，给出其Stinespring型分解，并刻画核支配关系，恢复纯态与不可约表示的经典对应。