A number of known results
on Jordan ∗-homomorphism between B∗-algebras are generalized to Jordan
∗-homomorphisms between reduced Banach *-algebras. However the main results
presented here are new even for maps between B∗-algebras. We state these
results briefly. For any ∗-algebra A, let AqU be the set of quasi-unitary
elements. Let A and B be reduced Banach ∗-algebras ( = A∗-algebras). Let
φ : A → B be a linear map. Then φ is a Jordan *-homomorphism if and only if
φ(AqU) ⊆ BqU. If φ is bijective these conditions are equivalent to φ being a weakly
positive isometry with respect to the Gelfand-Naimark norms of A and
B.
