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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 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.