| This dissertation is concerned with results describing the nature of derivations in a certain category. Known as {dollar}JBsp{lcub}*{rcub}{dollar}-triples it is a category of normed Jordan triple systems. This peculiar object of study is unusual in the sense that it is concerned not with the usual binary product which everyone is familiar with, but actually a ternary product. Although it seems foreign to people who have studied mathematics for a long time, there are familiar examples despite the fact that the product in these examples will no longer be binary.; Originally, {dollar}JBsp{lcub}*{rcub}{dollar}-triples occurred in the study of geometric objects known as bounded symmetric domains in finite and infinite dimensions. Kaup showed the equivalence of the two categories: bounded symmetric domains in complex Banach spaces and {dollar}JBsp{lcub}*{rcub}{dollar}-triples.; The motivation behind the study of these derivations instead of the usual binary ones is a theorem, again due to Kaup, which states that the ternary isomorphisms defined in terms of the ternary product, are precisely the surjective isometries. Consequently in this case, geometry is "equivalent" to algebra, and by studying these derivations we are studying the infinitesimal generator of one parameter group of isometries instead of just automorphisms.; Derivations, like many other operators can be either bounded or unbounded. In the first chapter of this dissertation, we shall give all the necessary background and definitions. Then we shall devote the second chapter to prove the result of Upmeier for JB-algebras in this category (Upmeier showed all derivations of a JBW-algebra are inner if and only if its spin representations have uniformly bounded dimensions). In chapter three we shall investigate the possibility of weak amenability for {dollar}JBsp{lcub}*{rcub}{dollar}-triples, which was done by Haagerup for associative algebras (in fact, Haagerup showed all nuclear {dollar}Csp{lcub}*{rcub}{dollar}-algebras are amenable in this paper, which is the converse of Connes' result). Of course these are chapters only for bounded derivations. We shall not consider unbounded derivations in this dissertation. |
