Quantum symmetries on operator algebras.

*(English)*Zbl 0924.46054
Oxford Mathematical Monographs. Oxford: Clarendon Press. xv, 829 p. (1998).

This is a most remarkable book and it certainly gives a hand to many mathematicians, researchers as well as graduate students. On less than 850 pages the authors present vital information about 124 subjects in 15 chapters, in a manner accessible to any broadly trained analyst. Non-specialists should not expect easy climbing from sea level but it can be done. As a collection of broad and relevant information the book bears some similarity to the famous volumes of Dunford and Schwartz.

After this appraisal, in order to give a flavour of the content of the book, it seems natural to close the review by quoting the titles to the 15 chapters (each of which is concluded by notes): 1) Operator theory basics 2) \(C^{*}\)-algebra basics 3) \(K\)-theory 4) Positivity and semigroups 5) von Neuman algebra basics 6) The Fermion algebra 7) The Ising model 8) Conformal field theory 9) Subfactors and bimodules 10) Axiomatization of paragroups 11) String algebras and flat connections 12) Topological quantum field theory and paragroups 13) Rational conformal field theory and paragroups 14) Commuting squares of II\(_1\)-factors 15) Authomorphisms of subfactors and central sequences. Curtain.

After this appraisal, in order to give a flavour of the content of the book, it seems natural to close the review by quoting the titles to the 15 chapters (each of which is concluded by notes): 1) Operator theory basics 2) \(C^{*}\)-algebra basics 3) \(K\)-theory 4) Positivity and semigroups 5) von Neuman algebra basics 6) The Fermion algebra 7) The Ising model 8) Conformal field theory 9) Subfactors and bimodules 10) Axiomatization of paragroups 11) String algebras and flat connections 12) Topological quantum field theory and paragroups 13) Rational conformal field theory and paragroups 14) Commuting squares of II\(_1\)-factors 15) Authomorphisms of subfactors and central sequences. Curtain.

Reviewer: W.Slowikowski (Aarhus)

##### MSC:

46L60 | Applications of selfadjoint operator algebras to physics |

46N50 | Applications of functional analysis in quantum physics |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

46L37 | Subfactors and their classification |

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

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |

82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |

81T25 | Quantum field theory on lattices |