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Book Information:
Authors:
D. Evans and
Y. Kawahigashi
Title:
Quantum symmetries on operator algebras
Additional book information:
Oxford Univ. Press,
New York,
1998,
xv + 829 pp.,
ISBN 0-19-851175-2,
$200.00$
[1] O. Bratteli, Inductive limits of finite dimensional C-algebras. Transactions AMS 171, (1972), 195-234. MR 0312282
[2]A. Connes, Classification of injective factors. Ann. of Math. 104, (1976), 73-115. MR 0454659
[3] A. Connes, Une classification des facteurs de type III. Ann. Sci. Ecole Norm Sup 6, (1973), 133-252. MR 0341115
[4]A. Connes, Outer conjugacy classes of automorphisms of factors. Annales Scientifiques de l'éc. norm. sup. 8, (1975), 383-419. MR 0394228
[5]S. Doplicher, R. Haag, J. Roberts, Local observables and particle statistics, I. Comm. Math. Phys. 23, (1971), 199-230. MR 0297259
[6] E. Effros, D. Handelman and C. Shen, Dimension groups and their affine representations. Amer. J. Math 102, (1980), 385-407. MR 0564479
[7] G. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensional algebras. Journal of Algebra 38, (1976), 29-44. MR 0397420
[8] K. Fredenhagen, K.-H. Rehren and B. Schroer, Superselection sectors with braid group statistics and exchange algebras. Comm. Math. Phys. 125, (1989), 201-226. MR 1016869
[9]J. Glimm, A. Jaffe, Quantum physics. A functional integral point of view. Springer-Verlag, New York-Berlin, 1981. MR 0628000
[10]F.M. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter graphs and towers of algebras, Springer-Verlag, 1989. MR 0999799
[11] R. Haag, Local quantum physics, Springer, Berlin, 1992. MR 1182152
[12] U. Haagerup, Principal graphs of subfactors in the index range in: ``Subfactors", World Scientific, Singapore-New Jersey-London-Hong Kong (1994) 1-39. MR 1317352
[13] -, Connes' bicentraliser problem and uniqueness of the injective factor of type III. Acta Math. 158, (1987), 95-148. MR 0880070
[14] V. Jones, Index for subrings of rings. Contemporary Math. 43, (1985), 181-190. MR 0810651
[15] R. Longo, Index of subfactors and statistics of quantum fields. I. Comm. Math. Phys. 126, (1989), 217-247. MR 1027496
[16] D. McDuff, Uncountably many II factors. Ann. of Math. 90, (1970), 372-377. MR 0259625
[17]F.J. Murray and J. von Neumann, On rings of operators. IV. Ann. of Math. 44, (1943), 716-808. MR 5:101a
[18] M. Nakamura, Z. Takeda, A Galois theory for finite factors. Proc. Japan Acad. 36, (1960), 258-260. MR 0123925
[19]A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras. Springer Lecture Notes in Mathematics, 1138, (1985). MR 0807949
[20] A. Ocneanu, Quantized group, string algebras and Galois theory for algebras in Operator algebras and applications, vol. 2, L.M.S lecture note series, 136, (1988), 119-172. MR 0996454
[21]S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math 120, (1995), 427-445. MR 1334479
[22]S. Sawin, Subfactors constructed from quantum groups. Amer. J. Math. 117, (1995), no. 6, 1349-1369. MR 1363071
[23]J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50, (1949), 401-485. MR 10:548a
[24]-, On rings of operators, III, Ann. of Math. 41, (1940), 94-161.
[25]-, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann. 102, (1929), 370-427.
[26]-, On infinite direct products, Compositio Math 6, (1938), 1-77.
[27]A. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of using bounded operators. Invent. Math. 133, (1998), 467-538. MR 1645078
[28]H. Wenzl, tensor categories from quantum groups. J. Amer. Math. Soc. 11, (1998), no. 2, 261-282. MR 1470857
[29]F. Xu, Standard -lattices from quantum groups. Invent. Math. 134, (1998), 455-487. MR 1660937
- [1]
- O. Bratteli, Inductive limits of finite dimensional C-algebras. Transactions AMS 171, (1972), 195-234. MR 0312282
- [2]
- A. Connes, Classification of injective factors. Ann. of Math. 104, (1976), 73-115. MR 0454659
- [3]
- A. Connes, Une classification des facteurs de type III. Ann. Sci. Ecole Norm Sup 6, (1973), 133-252. MR 0341115
- [4]
- A. Connes, Outer conjugacy classes of automorphisms of factors. Annales Scientifiques de l'éc. norm. sup. 8, (1975), 383-419. MR 0394228
- [5]
- S. Doplicher, R. Haag, J. Roberts, Local observables and particle statistics, I. Comm. Math. Phys. 23, (1971), 199-230. MR 0297259
- [6]
- E. Effros, D. Handelman and C. Shen, Dimension groups and their affine representations. Amer. J. Math 102, (1980), 385-407. MR 0564479
- [7]
- G. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensional algebras. Journal of Algebra 38, (1976), 29-44. MR 0397420
- [8]
- K. Fredenhagen, K.-H. Rehren and B. Schroer, Superselection sectors with braid group statistics and exchange algebras. Comm. Math. Phys. 125, (1989), 201-226. MR 1016869
- [9]
- J. Glimm, A. Jaffe, Quantum physics. A functional integral point of view. Springer-Verlag, New York-Berlin, 1981. MR 0628000
- [10]
- F.M. Goodman, P. de la Harpe, and V.F.R. Jones, Coxeter graphs and towers of algebras, Springer-Verlag, 1989. MR 0999799
- [11]
- R. Haag, Local quantum physics, Springer, Berlin, 1992. MR 1182152
- [12]
- U. Haagerup, Principal graphs of subfactors in the index range in: ``Subfactors", World Scientific, Singapore-New Jersey-London-Hong Kong (1994) 1-39. MR 1317352
- [13]
- -, Connes' bicentraliser problem and uniqueness of the injective factor of type III. Acta Math. 158, (1987), 95-148. MR 0880070
- [14]
- V. Jones, Index for subrings of rings. Contemporary Math. 43, (1985), 181-190. MR 0810651
- [15]
- R. Longo, Index of subfactors and statistics of quantum fields. I. Comm. Math. Phys. 126, (1989), 217-247. MR 1027496
- [16]
- D. McDuff, Uncountably many II factors. Ann. of Math. 90, (1970), 372-377. MR 0259625
- [17]
- F.J. Murray and J. von Neumann, On rings of operators. IV. Ann. of Math. 44, (1943), 716-808. MR 5:101a
- [18]
- M. Nakamura, Z. Takeda, A Galois theory for finite factors. Proc. Japan Acad. 36, (1960), 258-260. MR 0123925
- [19]
- A. Ocneanu, Actions of discrete amenable groups on von Neumann algebras. Springer Lecture Notes in Mathematics, 1138, (1985). MR 0807949
- [20]
- A. Ocneanu, Quantized group, string algebras and Galois theory for algebras in Operator algebras and applications, vol. 2, L.M.S lecture note series, 136, (1988), 119-172. MR 0996454
- [21]
- S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math 120, (1995), 427-445. MR 1334479
- [22]
- S. Sawin, Subfactors constructed from quantum groups. Amer. J. Math. 117, (1995), no. 6, 1349-1369. MR 1363071
- [23]
- J. von Neumann, On rings of operators. Reduction theory, Ann. of Math. 50, (1949), 401-485. MR 10:548a
- [24]
- -, On rings of operators, III, Ann. of Math. 41, (1940), 94-161.
- [25]
- -, Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren, Math. Ann. 102, (1929), 370-427.
- [26]
- -, On infinite direct products, Compositio Math 6, (1938), 1-77.
- [27]
- A. Wassermann, Operator algebras and conformal field theory. III. Fusion of positive energy representations of using bounded operators. Invent. Math. 133, (1998), 467-538. MR 1645078
- [28]
- H. Wenzl, tensor categories from quantum groups. J. Amer. Math. Soc. 11, (1998), no. 2, 261-282. MR 1470857
- [29]
- F. Xu, Standard -lattices from quantum groups. Invent. Math. 134, (1998), 455-487. MR 1660937
Review Information:
Reviewer:
Vaughan F. R. Jones
Affiliation:
University of California, Berkeley
Email:
vfr@math.berkeley.edu
Journal:
Bull. Amer. Math. Soc.
38 (2001), 369-377
Published electronically:
March 27, 2001
Review copyright:
© Copyright 2001
American Mathematical Society