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**Global analysis on foliated spaces.**
*(English)*
Zbl 0648.58034

Mathematical Sciences Research Institute Publications, 9. New York etc.: Springer-Verlag. 337 p. DM 72.00 (1988).

This book grew out of lectures given by the authors at Berkeley (C. C. Moore) and at UCLA (C. Schochet), respectively. Its purpose is to develop a variety of aspects of analysis and geometry on foliated spaces. The aim is to give an exposition to A. Connes’ index theorem on foliated spaces [A. Connes, Lect. Notes Math. 725, 19-143 (1979; Zbl 0412.46053)] and to give background material of the index theorem of A. Connes and G. Skandalis [C. R. Acad. Sci., Paris, Sér. I 292, 871-876 (1981; Zbl 0529.58030)].

The table of contents is the following: Introduction (very readable), I. Locally traceable operators, II. Foliated spaces, III. Tangential cohomology, IV. Transverse measures, V. Characteristic classes, VI. Operator algebras, VII. Pseudodifferential operators, VIII. The index theorem. The book ends with three appendices. The first written by S. Hurder: The \({\bar \partial}\) operator. The second is written by the authors jointly with R. J. Zimmer and is concerned with: \(L^ 2\)- harmonic forms on non-compact manifolds. The last appendix is due to R. J. Zimmer and titled: Positive scalar curvature along leaves.

The table of contents is the following: Introduction (very readable), I. Locally traceable operators, II. Foliated spaces, III. Tangential cohomology, IV. Transverse measures, V. Characteristic classes, VI. Operator algebras, VII. Pseudodifferential operators, VIII. The index theorem. The book ends with three appendices. The first written by S. Hurder: The \({\bar \partial}\) operator. The second is written by the authors jointly with R. J. Zimmer and is concerned with: \(L^ 2\)- harmonic forms on non-compact manifolds. The last appendix is due to R. J. Zimmer and titled: Positive scalar curvature along leaves.

Reviewer: N.Jacob

### MSC:

58J20 | Index theory and related fixed-point theorems on manifolds |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

57R30 | Foliations in differential topology; geometric theory |

46L05 | General theory of \(C^*\)-algebras |

28D15 | General groups of measure-preserving transformations |