We classify certain algebras of matrix-valued cross-sections over an annulus up to complete isometric isomorphism, based on topological bundle invariants. In particular, we study sections of matrix bundles which are continuous on the closure of the annulus and holomorphic on its interior. Our strategy includes exploiting the relationship between concomitants and modulus automorphic functions, as well as the classification of n-homogeneous C*-algebras by Fell and Tomiyama-Takesaki. Furthermore, we describe a partial extension of our results over the annulus to larger classes of finitely and smoothly bordered planar domains.

Part of this work began at the University of Iowa as a part of the second-named author's PhD thesis. The first-named author was supported by a California State University Long Beach student research assistantship fellowship made possible by Richard D. Green funding during the summer and fall of 2021.