Finite dimensional quotients of commutative operator algebras

TL;DR

This paper classifies finite-dimensional quotients of commutative operator algebras, explores their structure, and compares their properties with those of function algebra quotients, highlighting new counterexamples and generalizations of classical theories.

Contribution

It provides a classification of two-dimensional unital operator algebras, introduces invariant distances, and generalizes Nevanlinna-Pick theory for quotients of the d-shift algebra.

Findings

Classification of 2D unital operator algebras

Counterexamples with badly behaved quotients

Explicit isometric representations of quotients of the d-shift

Abstract

The matrix normed structure of the unitization of a (non-selfadjoint) operator algebra is determined by that of the original operator algebra. This yields a classification up to completely isometric isomorphism of two-dimensional unital operator algebras. This allows to define invariant distances on the spectrum of commutative operator algebras analogous to the Caratheodory distance for complex manifolds. Moreover, unitizations of two-dimensional operator algebras with zero multiplication provide a rich class of counterexamples. Especially, several badly behaved quotients of function algebras are exhibited. Recently, Arveson has developed a model theory for d-contractions. Quotients of the operator algebra of the d-shift are much more well-behaved than quotients of function algebras. Completely isometric representations of these quotients are obtained explicitly. This provides a generalization of Nevanlinna-Pick theory. An important property of quotients of the d-shift algebra is that their quotients of finite dimension r have completely isometric representations by rxr-matrices. Finally, the class of commutative operator algebras with this property is investigated.