All algebras considered
are complex Banach algebras with identity and continuous involution. The
principal results of §1 are that for a Jordan ∗homomorphism T of A1 into
A2 where A2 is ∗semisimple, continuity is automatic, the kernel is a closed
∗ideal, and if A2 is commutative then the factor algebra A1∕kernel T is also
commutative. In §2 a cone different from the usual cone is introduced and its
relation to the usual cone is studied. The principal result is that if this cone
coincides with the usual cone, then any Jordan ∗representation is the sum of a
∗representation and a ∗antirepresentation. §3 is devoted to proving that for a
∗semisimple algebra, the axiom ∥xy∥≦∥x∥∥y∥ follows from the weaker axiom
∥xy + yx∥≦ 2∥x∥∥y∥.
