Operator Algebras of Functions

TL;DR

This paper develops a unifying operator algebra framework for function theories on domains, extending existing theories and providing new factorization and duality results for operator algebras.

Contribution

It introduces general theorems on operator algebras of functions, unifies various domain theories, and extends classical results with new factorization and duality theorems.

Findings

Established that certain operator algebras are dual operator algebras.

Proved supremum equivalences over operator tuples and matrices.

Extended classical function theory to a broader operator algebra context.

Abstract

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 dual operator algebras and show that for many domains, supremums over all commuting tuples of operators satisfying certain inequalities are obtained over all commuting tuples of matrices.