Operator Theory Seminar
There is a long tradition of analyzing a $C^*$-algebra through some topological invariant. One such result is Tomiyama and Takesaki's 1961 proof that an $n$-homogeneous $C^*$-algebra $A$ is determined up to $*$-isomorphism by an underlying continuous matrix bundle, and $A$ is an algebra of continuous cross-sections of the bundle. Suppose that the base space of the bundle is a bordered Riemann surface with finitely many smooth boundary components. Then for each such $n$-homogeneous $C^*$-algebra, one can define a subalgebra of holomorphic cross-sections. We will describe a partial result towards classifying these subalgebras up to complete isometric isomorphism based on topological invariants.