×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI