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We classify certain algebras of matrix-valued cross-sections over an annulusup to complete isometric isomorphism, based on topological bundle invariants.In particular, we study sections of matrix bundles which are continuous on theclosure of the annulus and holomorphic on its interior. Our strategy includesexploiting the relationship between concomitants and modulus automorphicfunctions, as well as the classification of $n$-homogeneous $C^*$-algebras byFell and Tomiyama-Takesaki. Furthermore, we describe a partial extension of ourresults over the annulus to larger classes of finitely and smoothly borderedplanar domains.

Classifying Matrix-Valued Holomorphic Cross-Sections Over An Annulus Up To Complete Isometric Isomorphism