Prime ideals in C*-algebras and applications to Lie theory

We show that every proper, dense ideal in aC∗C*-algebra is contained in a prime ideal. It follows that a subset generates aC∗C*-algebra as a not necessarily closed ideal if and only if it is not contained in any prime ideal. This allows us to transfer Lie theory results from prime rings toC∗C*-algebras. For example, if aC∗C*-algebraAAis generated by its commutator subspace[A,A][A,A]as a ring, then[[A,A],[A,A]]=[A,A][[A,A],[A,A]] = [A,A]. Further, given Lie idealsKKandLLinAA, then[K,L][K,L]generatesAAas a not necessarily closed ideal if and only if[K,K][K,K]and[L,L][L,L]do, and moreover this implies that[K,L]=[A,A][K,L]=[A,A]. We also discover new properties of the subspace generated by square-zero elements and relate it to the commutator subspace of aC∗C*-algebra.