Jordan *-derivations and quadratic functionals on octonion algebras
作者:
Borut Zalar,
期刊:
Communications in Algebra
(Taylor Available online 1994)
卷期:
Volume 22,
issue 8
页码: 2845-2859
ISSN:0092-7872
年代: 1994
DOI:10.1080/00927879408824996
出版商: Marcel Dekker, Inc.
数据来源: Taylor
摘要:
Let A be a *-algebra. An additive mapping E : A → A is called a Jordan *-derivation if E(X2) = E(x)x*+xE(x) holds, for all x 6 A. These mappings have been extensively studied in the last 6 years by Bresar, Semrl, Vukman and Zalar because they are closely connected with the problem of representability of quadratic functionals by sesquilinear forms. This study was, however, always in the setting of associative rings. In the present paper we study Jordan *-derivations on the Cayley-Dickson algebra of octonions, which is not associative. Our first main result is that every Jordan *-derivation on the octonion algebra is of the form E(x)=ax*-xa. In the terminology of earlier papers this means that every Jordan *-derivation on the octonion algebra is inner. This generalizes the known fact that Jordan *-derivations on complex and quaternion algebras are inner. Our second main result is a representation theorem for quadratic functionals on octonion modules. Its proof uses the result mentioned above on Jordan *-derivations.
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