Quantum symmetries on operator algebras. (English) Zbl 0924.46054

Evans, David E.; Kawahigashi, Yasuyuki

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.

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

Keywords:

operator theory; \(C^*\)-algebra; \(K\)-theory; positivity; semigroups; von Neumann algebra; fermion algebra; Ising model; conformal field theory; subfactors; bimodules; paragroups; string algebras; flat connections; topological quantum field theory; rational conformal field theory; automorphisms; central sequences