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 von~Neumann inequality. The new ingredient is a certain set of matrix valued functions termed "auxiliary test functions" used in constructing transfer functions. In important ses, the realization theorems can be strengthened so as to allow applications to Pick type interpolation problems, among other things. Principle examples have as the domain the polydisk $\mathbb D^d$. The algebras then include $H^\infty(\mathbb D^d,\mathcal{L(H)})$ (where the unit ball is traditionally called the Schur class) and $A(\mathbb D^d,\mathcal{L(H)})$, the multivariable analogue of the disk algebra. As an application, it is shown that over the polydisk $\mathbb D^d$, (weakly continuous) representations which are $2^{d-2}$ contractive are completely contractive (hence having a commuting unitary dilation), offering fresh insight into such examples as Parrott's of contractive representations of $A(\mathbb D^3)$ which are not completely contractive.
Test Functions, Kernels, Realizations and Interpolation
Jim Agler用他的实现定理彻底改变了Pick插值的领域，该定理现在被称为$\mathbb C^d$中单位球的Agler-Schur类。我们讨论了这些结果所产生的检验函数代数和对偶概念的家庭再生核，以及相关的内插定理。在使用检验函数时，理想情况下希望使用尽可能小的集合。然而，在某些情况下，检验函数的无限集合是不可避免的。在这种情况下，某些拓扑考虑就显得尤为重要。我们用例子来说明这一点，包括环形和无限多圆盘的乘子代数。
Schur类，用$\mathcal{S}(\mathbb{D})$表示，是所有解析函数的集合，并且以复平面$\mathbb{C}$中的开单位圆盘$\mathbb{D}$中的模为1为界，即\[ \mathcal{S}(\mathbb{D}) = \{φ\in H^\infty(\mathbb{D}): \|φ\|_{\infty} := \sup_{z \in \mathbb{D}} |φ(z)| \leq 1\}. \]。$\mathcal{S}(\mathbb{D})$的元素称为Schur函数。一个经典的结果可以追溯到I。Schur指出：一个函数$φ: \mathbb{D} \rightarrow \mathbb{C}$在$\mathcal{S}(\mathbb{D})$中当且仅当存在一个希尔伯特空间$\mathcal{H}$和一个等距（称为结合算子矩阵或散射算子矩阵）\[ V = \begin{bmatrix} a & B \\ C & D \end{bmatrix} : \mathbb{C} \oplus \mathcal{H} \rightarrow \mathbb{C} \oplus \mathcal{H}, \]，使得$φ$允许一个对应于$V$的传递函数实现，即\[ φ(z) = a + z B (I_{\mathcal{H}} - z D)^{-1} C \quad …
Function theory of the hexablock and applications to the tetrablock and Euclidean biball
Mathematics Department, Indian Institute of Technology Bombay（印度理工学院孟买分校数学系）；Statistics and Mathematics unit, Indian Statistical Institute Bangalore（印度统计学院班加罗尔分校统计与数学单元）
