On $n$-homogeneous $C^*$-algebras over a two-dimensional compact oriented connected manifold

We consider the $n$-homogeneous $C^*$-algebras over a two-dimensional compact oriented connected manifold. Suppose $A$ be the $n$-homogeneous $C^*$-algebra with space of primitive ideals homeomorphic to a two-dimensional connected oriented compact manifold $P(A)$. It is well known that the manifold $P(A)$ is homeomorphic to the sphere $P_k$ glued together with $k$ handles in the hull-kernel topology. On the other hand, the algebra $A$ is isomorphic to the algebra $\Gamma (E)$ of continuous sections for the appropriate algebraic bundle $E$. The base space for the algebraic bundle is homeomorphic to the set $P_k$. By using this geometric realization, we described the class of non-isomorphic $n$-homogeneous ($n\geq 2$) $C^*$-algebras over the set $P_k$. Also, we calculated the number of non-isomorphic $n$-homogeneous $C^*$-algebras over the set $P_k$.