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Function representation of commutative operator triple systems. (English) Zbl 0543.46046
An operator triple system means a complex linear subspace of $${\mathcal L}(H,K)$$, the bounded linear operators from a Hilbert space H to a Hilbert space K, which is closed in some topology and under some triple product $$\{$$.,.,.$$\}$$ of its elements. Examples of operator triple systems are given by $$J^*$$-algebras, norm closed subspaces of $${\mathcal L}(H,K)$$ which contain $$aa^*a$$ along with the element a; this is because a $$J^*$$-algebra is closed under the triple product $$\{a,b,c\}=ab^*c+cb^*a.$$ This paper deals with triple systems which satisfy the identities $\{xy\{zuv\}\}=\{x\{uzy\}v\}=\{\{xyz\}uv\},$ i.e. associative Jordan triple systems. In particular if a $$J^*$$- algebra becomes an associative Jordan triple system in the triple product $$\{xyz\}=1/2(xy^*z+zy^*x)$$ then it is said to be a commutative $$J^*$$-algebra.
The main result of the paper (Theorem 1) states that a commutative $$J^*$$-algebra is isometrically $$J^*$$-isomorphic to a space of all continuous complex functions satisfying some symmetry properties. One of the principal consequences of the main result is a Gelfand-Neumark representation theorem for associative Jordan triple systems (Theorem 2). There are also some applications of Theorem 1 to the study of contractive projections, Banach-Stone type theorems, Stone-Weierstrass theorems in the setting of commutative $$J^*$$-algebras.
Reviewer: Sh.A.Ayupov

##### MSC:
 46L99 Selfadjoint operator algebras ($$C^*$$-algebras, von Neumann ($$W^*$$-) algebras, etc.) 46J99 Commutative Banach algebras and commutative topological algebras 17C65 Jordan structures on Banach spaces and algebras 46J10 Banach algebras of continuous functions, function algebras 46J30 Subalgebras of commutative topological algebras
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