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**Cônes autopolaires et algèbres de Jordan.**
*(French)*
Zbl 0556.46040

Lecture Notes in Mathematics. 1049. Berlin etc.: Springer-Verlag. VI, 247 p. DM 33.50; $ 12.50 (1984).

This book is primarily devoted to the study of selfdual cones associated with J.B.W. algebras. it presents a fairly complete discussion of these cones, profiting greatly from the developments of the theory of von Neumann algebras and ordered topological vector spaces that took place in the last decade or so.

Homogeneous selfdual cones (finite dimensional !) have first been investigated and classified by Koecher. He also established the relation between cones and formally real Jordan algebras. Several authors have generalized the correspondence between cones and Jordan algebras to certain infinite dimensional situations. The first fairly general result relating cones to algebras in infinite dimensions was proven by Connes. He set up a correspondence between von Neumann algebras and ”orientable”, ”facially homogeneous” selfdual cones in complex Hilbert spaces. A cone is called ”facially homogeneous” if for a face F of the cone the difference of projections \(P_ F-P_{F^{\perp}}\) generates a one- parameter group of automorphisms of the cone. (For the definition of ”orientable” we refer to chapter VI.)

Thus one can formulate the main result of the book:

The category of Jordan-Banach algebras with predual together with linear order preserving isomorphisms (resp. Jordan isomorphisms) is isomorphic to the category of facially homogeneous selfdual cones in real Hilbert spaces together with linear bijections (resp. linear unitary maps) preserving the order.

To prove this result and to illustrate this correspondence the author presents a full fledged investigation of (facially homogeneous) selfdual cones and their associated algebras.

Here is the list of the chapters: I. Selfdual Cones, II. Facially Homogeneous Selfdual Cones, III. The Jordan Algebra of a Facially Homogeneous Selfdual Cone, IV. Weights of a Selfdual Cone, V. Traces of J.B. Algebras, VI. Orientable Cones, VII. Cones associated to J.B.W. Algebras, Appendices.

This book is well written. The author clearly makes an effort to explain the development of the theory and to give proper credit. This makes the introduction to the book and the separate introductions to the various chapters particularly interesting and helpful. I would have preferred though to see the (too) nany appendices beig incorporated into the main part of the text.

Homogeneous selfdual cones (finite dimensional !) have first been investigated and classified by Koecher. He also established the relation between cones and formally real Jordan algebras. Several authors have generalized the correspondence between cones and Jordan algebras to certain infinite dimensional situations. The first fairly general result relating cones to algebras in infinite dimensions was proven by Connes. He set up a correspondence between von Neumann algebras and ”orientable”, ”facially homogeneous” selfdual cones in complex Hilbert spaces. A cone is called ”facially homogeneous” if for a face F of the cone the difference of projections \(P_ F-P_{F^{\perp}}\) generates a one- parameter group of automorphisms of the cone. (For the definition of ”orientable” we refer to chapter VI.)

Thus one can formulate the main result of the book:

The category of Jordan-Banach algebras with predual together with linear order preserving isomorphisms (resp. Jordan isomorphisms) is isomorphic to the category of facially homogeneous selfdual cones in real Hilbert spaces together with linear bijections (resp. linear unitary maps) preserving the order.

To prove this result and to illustrate this correspondence the author presents a full fledged investigation of (facially homogeneous) selfdual cones and their associated algebras.

Here is the list of the chapters: I. Selfdual Cones, II. Facially Homogeneous Selfdual Cones, III. The Jordan Algebra of a Facially Homogeneous Selfdual Cone, IV. Weights of a Selfdual Cone, V. Traces of J.B. Algebras, VI. Orientable Cones, VII. Cones associated to J.B.W. Algebras, Appendices.

This book is well written. The author clearly makes an effort to explain the development of the theory and to give proper credit. This makes the introduction to the book and the separate introductions to the various chapters particularly interesting and helpful. I would have preferred though to see the (too) nany appendices beig incorporated into the main part of the text.

Reviewer: J.Dorfmeister

### MSC:

46L99 | Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.) |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

46A40 | Ordered topological linear spaces, vector lattices |

17C65 | Jordan structures on Banach spaces and algebras |

46H99 | Topological algebras, normed rings and algebras, Banach algebras |

17C50 | Jordan structures associated with other structures |

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |

06F25 | Ordered rings, algebras, modules |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |