On $3$-homogeneous $C^{*}$-algebras over two-dimensional oriented manifolds

We consider algebraic bundles over a two-dimensional compact oriented connected manifold. In 1961 J. Fell, J. Tomiyama, M. Takesaki showed that every $n$-homogeneous $C^{*}$-algebra is isomorphic to the algebra of all continuous sections for the appropriate algebraic bundle. By using this realization we prove in the work that every $3$-homogeneous $C^{*}$-algebra over two-dimensional compact oriented connected manifold can be generated by three idempotents. Such algebra can not be generated by two idempotents.