Function representation of commutative operator triple systems.

*(English)*Zbl 0543.46046An 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.

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 |