Universal commutative operator algebras and transfer function realizations of polynomials

Abstract

To each finite-dimensional operator space $E$ is associated a commutative operator algebra $UC(E)$, so that $E$ embeds completely isometrically in $UC(E)$ and any completely contractive map from $E$ to bounded operators on Hilbert space extends uniquely to a completely contractive homomorphism out of $UC(E)$. The unit ball of $UC(E)$ is characterized by a Nevanlinna factorization and transfer function realization. Examples related to multivariable von Neumann inequalities are discussed.