Matrix Bundles and Operator Algebras Over a Finitely Bordered Riemann Surface

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.