Jordan Homomorphisms and Derivations on Semisimple Banach Algebras
نویسنده
چکیده
1. Introduction. One may construct a Jordan homomorphism from one (associative) ring into another ring by taking the sum of a homo-morphism and an antihomomorphism of the first ring into two ideals in the second ring with null intersection [6]. A number of authors have considered conditions on the rings that imply that every Jordan homomorphism, or isomorphism, is of this form [6], [3], [7], [10], [ll]. We use a theorem of I. N. Herstein [3] (and without restrictions on the characteristic of the ring [lO]) to show that a Jordan homo-morphism from a ring onto a semisimple ring with identity and no one-dimensional irreducible (left) modules is the sum of a homomor-phism and an antihomomorphism (Theorem 2.1). The ideals with null intersection are obtained from a disconnection of the structure space in a way similar to that used in the proof of Corollary 3.8 of [ll, p. 446]. I am indebted to B. E. Johnson for drawing my attention to [ll]. We give an example to show that the conclusion is false if the hypothesis that there be no one-dimensional irreducible modules is omitted. In Theorem 2.2 we show that for C*-algebras the decomposition of Theorem 2.1 holds if the assumption that there is an identity is dropped. 1. N. Herstein has shown that a Jordan derivation on a prime ring not of characteristic 2 is a derivation [4]. We use this result to show that a continuous Jordan derivation on a semisimple Banach algebra
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