On $C^{*}$-algebras generated by idempotents

Banach algebras generated by two idempotents appear in many places. In 1968–1969 P. R. Halmos and G. K. Pedersen studied $C^{*}$-algebras generated by two self-adjoint projections. The Banach algebras generated by two idempotents were described by S. Roch and B. Silbermann in 1988. Such algebras can have irreducible representations of first or second order. The theory of Banach algebras generated by three idempotents has not yet been constructed. Such algebras can have irreducible representations of any order. In 1974 F. Krauss and T. Lawson described the $n$-homogeneous $C^{*}$-algebras over spheres $S^{2}, S^{3}, S^{4}$. By using these results we prove that $n$-homogeneous ($n>2$) $C^{*}$-algebra such that $PrimA\cong S^{4}$ can be generated by finite number of idempotents.