CONTINUITY OF JORDAN *-HOMOMORPHISMS OF BANACH *-ALGEBRAS

In this note we prove the following result: Let A be a complex Banach *-algebra with continuous involution and let B be an -algebra./T(A) = B. Then T is continuous (Theorem 2). From above theorem some others results of special interest and some well-known results follow. (Corollaries 3,4,5,6 and 7). We close this note with some generalizations and some remarks (Theorems 8.9.10 and question). Throughout this note we consider only complex algebras. Let A and B be complex algebras. A linear mapping T from A into B is called jordan homomorphism if T( ) = (Tx) for all x in A. A linear mapping T : A .rarw. B is called spectrally-contractive mapping if .rho.(Tx).leq..rho.(x) for all x in A, where .rho.(x) denotes spectral radius of element x. Any homomorphism algebra is a spectrally-contractive mapping. If A and B are *-algebras, then a homomorphism T : A.rarw.B is called *-homomorphism if (Th)=Th for all self-adjoint element h in A. Recall that a Banach *-algebras is a complex Banach algebra with an involution *. An -algebra A is a Banach *-algebra having anauxiliary norm vertical bar . vertical bar which satisfies -condition vertical bar x vertical bar = vertical bar x vertical ba (x in A). A Banach *-algebra whose norm is an algebra -norm is called -algebra. The *-semi-simple Banach *-algebras and the semi-simple hermitian Banach *-algebras are -algebras. Also, -algebras include -algebras ( -algebras). Recall that a semi-prime algebra is an algebra without nilpotents two-sided ideals non-zero. The class of semi-prime algebras includes the class of semi-prime algebras and the class of prime algebras. For all concepts and basic facts about Banach algebras we refer to [2] and [8].].er to [2] and [8].].