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

Abstract

We show that every proper, dense ideal in a C ∗ C^{*} -algebra is contained in a prime ideal. It follows that a subset generates a C ∗ 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 to C ∗ C^{*} -algebras. For example, if a C ∗ C^{*} -algebra A A is 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 ideals K K and L L in A A , then [ K , L ] [K,L] generates A A as 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 a C ∗ C^{*} -algebra.