We first consider the theory of
Jordan homomorphisms and Jordan ideals in Banach algebras. If B is a B∗-algebra
or a semi-simple annihilator algebra, any closed Jordan ideal in B is a two-sided
ideal. Any Jordan homomorphism of a Banach algebra onto B is automatically
continuous. That Jordan homomorphisms are continuous and Jordan ideals are ideals
is shown to hold in a number of other situations. We also study the Lie ideals in a
semi-simple Banach algebra A. If the center of A is zero and proper closed Lie ideals
do not contain their Lie annihilators, then A is direct topological sum of its
minimal closed ideals. An H∗-algebra with zero center is an example of such an
algebra.
