Matrix Bundles and Operator Algebras Over a Finitely Bordered Riemann Surface

TL;DR

This paper investigates operator algebras derived from flat holomorphic matrix bundles over finitely bordered Riemann surfaces, analyzing their boundary representations and establishing their Azumaya algebra structure.

Contribution

It provides a detailed analysis of boundary representations and proves that these operator algebras are Azumaya, connecting complex geometry with operator algebra theory.

Findings

Boundary representations correspond to evaluations on the Riemann surface boundary.

The operator algebras are shown to be Azumaya algebras.

The study extends understanding of bundle shift algebras in complex analysis.

Abstract

This note presents an analysis of a class of operator algebras constructed as cross-sectional algebras of flat holomorphic matrix bundles over a finitely bordered Riemann surface. These algebras are partly inspired by the bundle shifts of Abrahamse and Douglas. The first objective is to understand the boundary representations of the containing $C^*$-algebra, i.e. Arveson's noncommutative Choquet boundary for each of our operator algebras. The boundary representations of our operator algebras for their containing $C^*$-algebras are calculated, and it is shown that they correspond to evaluations on the boundary of the Riemann surface. Secondly, we show that our algebras are Azumaya algebras, the algebraic analogues of $n$-homogeneous $C^*$-algebras.