PRIME IDEALS IN C*-ALGEBRAS AND APPLICATIONS TO LIE THEORY

Journal article, 2024

. We show that every proper, dense ideal in a C & lowast;-algebra is contained in a prime ideal. It follows that a subset generates a C & lowast;-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 & lowast;-algebras. For example, if a C & lowast;-algebra A is generated by its commutator subspace [A, A] as a ring, then [[A, A], [A, A]] = [A, A]. Further, given Lie ideals K and L in A, then [K, L] generates A as a not necessarily closed ideal if and only if [K, K] and [L, L] do, and moreover this implies that [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 & lowast;-algebra.