## Ternary structures in Hilbert spaces

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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 JH-triple if the identity hfa; b; xg; xi =
hx; fb; a; xgi holds in V . JH-triples 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
Hilbert-Schmidt 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 anti-isomorphism is of the form (x) = Jx j where j : H0 ! K and
J : H ! K0 are isometries.
The structures of Hilbert ternary algebras and JH-triples are closely related.
Indeed, we show that each JH-triple (V; f ; ; g) admits a decomposi-
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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 JH-triples 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 JH-triple
(V; f ; ; g) for which V 6= Vs, equivalently, (V; f ; ; g) is not a Hilbert ternary
algebra.
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##### Authors

Bahmani, Fatemeh##### Collections

- Theses [2764]