Title:

Ternary structures in Hilbert spaces

Ternary structures in Hilbert spaces arose in the study of in nite dimensional manifolds in di erential geometry. In this thesis, we develop a structure theory of Hilbert ternary algebras and Jordan Hilbert triples which are Hilbert spaces equipped with a ternary product. We obtain several new results on the classi  cation of these structures. Some results have been published in [2]. A Hilbert ternary algebra is a real Hilbert space (V; h ; i) equipped with a ternary product [ ; ; ] satisfying h[a; b; x]; yi = hx; [b; a; y]i for a; b; x and y in V . A Jordan Hilbert triple is a real Hilbert space in which the ternary product f ; ; g is a Jordan triple product. It is called a JHtriple if the identity hfa; b; xg; xi = hx; fb; a; xgi holds in V . JHtriples correspond to a class of Lie algebras which play an important role in symmetric Riemannian manifolds. We begin by proving new structure results on ideals, centralizers and derivations of Hilbert ternary algebras. We characterize primitive tripotents in Hilbert ternary algebras and use them as coordinates to classify abelian Hilbert ternary algebras. We show that they are direct sums of simple ones, and each simple abelian Hilbert ternary algebra is ternary isomorphic to the algebra C2(H;K) of HilbertSchmidt operators between real, complex or quaternion Hilbert spaces H and K. Further, we describe completely the ternary isomorphisms and ternary antiisomorphisms between abelian Hilbert ternary algebras. We show that each ternary isomorphism between simple algebras C2(H;K) and C2(H0;K0) is of the form (x) = Jxj where j : H0 ! H and J : K ! K0 are isometries. A ternary antiisomorphism is of the form (x) = Jx j where j : H0 ! K and J : H ! K0 are isometries. The structures of Hilbert ternary algebras and JHtriples are closely related. Indeed, we show that each JHtriple (V; f ; ; g) admits a decomposi 6 tion V = Vs L V ? s where (Vs; f ; ; g) is a Hilbert ternary algebra which is usually nonabelian and unless V = Vs, the orthogonal complement V ? s is always a nonabelian Hilbert ternary algebra in the derived ternary product [a; b; c]d = fa; b; cg fb; a; cg. Hence JHtriples provide important examples of nonabelian Hilbert ternary algebras. We determine exactly when Vs and V ? s are Jordan triple ideals of V . We show, in each dimension at least 2, there is a JHtriple (V; f ; ; g) for which V 6= Vs, equivalently, (V; f ; ; g) is not a Hilbert ternary algebra. 7
