Derivações multiplicativas de Jordan sobre anéis com involução

Abstract

The study of Jordan derivations has been motivated by the representation problem for bilinear and quadratic forms. More precisely, the question if a quadratic form can be represented by a bilinear form is connected with the structure of *-derivations, as has been shown by ˇSemrl [23]. Many results about Jordan *-derivations are mentioned in literature and motivated the study of the following question: in a noncommutative prime ring R with involution, any *-derivation is X -inner. In the first part of the dissertation, we present a proof of this result for a ring R with characteristic not 2, published by Lee and Zhou [17] in 2014. For this, we used as a tool the theory of functional identities, introduced in Breˇsar [5] thesis, in 1990, whose general foundations established by Beidar [2] in the 90’s, and has connections with many areas, as Mathematical Physics, Functional Analysis, Operator Theory, Linear Algebra, Jordan Algebras, Lie Algebras and other non-associative algebras. On the other hand, the definition of Jordan *-derivation does not assume does not assume linearity or additivity. So, one natural and interesting question is to determinate under which hypothesis a Jordan *-derivation is additive. This was described by Qi and Zhang [20] in 2016. In the second part of the dissertation, our goal is to present in detail the proofs of theorems concerning the above question. At first, we characterize Jordan multiplicative *-derivations under the action of zero products, and secondly, we take off this last hypothesis and study the general case. Finally, we present some consequences, among which one concrete description of Jordan *-derivations over noncommutative prime *-rings. This generalizes already known results about these applications.