On trivial and non-trivial n-homogeneousC* algebras

LetA be aC* — algebra for which all irrèducible representations are of dimensional n. Then ([F], [TT], [V]) algebraA is isomorphic to algebra of all continuous sections of an appropriate algebraic bundle εA. The basisX of this bundle coincides with the compact of all maximal two-sided ideals ofA. We obtain some conditions which provide that εA is trivial and this yields thatA is isomorphic to the algebra of alln×n matrix functions continuous onX. In the case whenX=Sn is a sphere we describe the set of algebraic bundles overX and algebraic structures on this set. Some applications to algebras generated by idempotents are suggested.