Non-commutative Caratheodory Interpolation

TL;DR

This paper extends classical Caratheodory-Fejer interpolation to matrix convex sets in multiple dimensions using operator algebra techniques, generalizing previous work on non-commutative polydiscs.

Contribution

It introduces a new interpolation theorem for matrix convex sets in several variables, broadening the scope of non-commutative function theory.

Findings

Proves a non-commutative Caratheodory-Fejer interpolation theorem.

Utilizes Blecher-Ruan-Sinclair characterization of operator algebras.

Generalizes previous results on non-commutative polydiscs.

Abstract

We prove a Caratheodory-Fejer type interpolation theorem for certain matrix convex sets in $\C^d$ using the Blecher-Ruan-Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhnyi-Verbovetzkii for the d-dimensional non-commutative polydisc.