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**Von Neumann algebras in mathematics and physics.**
*(English)*
Zbl 0781.46046

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 121-138 (1991).

[For the entire collection see Zbl 0741.00019.]

This survey touches a series of topics initiated by the author’s study of von Neumann factors. The summary of well known concepts of von Neumann algebras includes Tomita-Takesaki theory, the classification of hyperfinite von Neumann algebras and the author’s work on index for subfactors leading to the basic construction of towers of subfactors of a \(\text{II}_ 1\) factor and the corresponding system of orthogonal projections. This system of projections is the crux of almost all the discussed relationships between operator algebras and the other fields of mathematics and physics which are described here. The author touches on the commuting square condition and the combinatorics for centraliser towers of finite-dimensional algebras arising from factors \(N\subset M\), \(M\) of type \(\text{II}_ 1\), and the classification of finite index subfactors of the hyperfinite \(\text{II}_ 1\) factor, or finite depth [see e.g., F. Goodman, P. de la Harpe and V. Jones, Coxeter graphs and towers of algebras (1989; Zbl 0698.46050)]. The related fields include braid groups on strings (which feature in quantum groups and solvable models of statistical mechanics, and thread through the entire article), Hecke algebras (complex algebras with generators related to the system of projection operators) and statistical mechanical models [cf. H. V. N. Temperley and E. H. Lieb, Proc. R. Soc. Lond., Ser. A 322, 251-280 (1971; Zbl 0211.567) who deal with tensor product representations of the projection systems]. Further developments dealt with the knot polynomials as introduced by the author in Bull. Am. Math. Soc., New Ser. 12, 103-111 (1985; Zbl 0564.57006) (New invariants for knots and links), positivity of the Markov trace (a special trace leading to \(\text{II}_ 1\) factors) on Hecke algebras [see H. Wenzl, Invent. Math. 92, No. 2, 349-383 (1988; Zbl 0663.46055)], the BMW algebra [J. Murakami, Osaka J. Math. 24, 745-758 (1987; Zbl 0666.57006), J. S. Birman and H. Wenzl, Trans. Am. Math. Soc. 313, No. 1, 249-273 (1989; Zbl 0684.57004)] and the connection of braid group representations of factors to conformal field theory. The author suggests a connection between braid group representations and algebraic quantum field theory (and in particular superselection sectors) and loop groups. The author gives some conjectures on the connection to loop groups via vertex operators [A. Tsuchiya and Y. Kanie, Adv. Stud. Pure Math. 16, 297-372 (1988; Zbl 0661.17021)] and to topological field theory. He also describes E. Wittens ideas on topological quantum field theory [Commun. Math. Phys. 121, No. 3, 351-399 (1989; Zbl 0667.57005)] and M. Atiyah [Publ. Math., IHES 68, 175-186 (1988; Zbl 0692.53053)].

This survey touches a series of topics initiated by the author’s study of von Neumann factors. The summary of well known concepts of von Neumann algebras includes Tomita-Takesaki theory, the classification of hyperfinite von Neumann algebras and the author’s work on index for subfactors leading to the basic construction of towers of subfactors of a \(\text{II}_ 1\) factor and the corresponding system of orthogonal projections. This system of projections is the crux of almost all the discussed relationships between operator algebras and the other fields of mathematics and physics which are described here. The author touches on the commuting square condition and the combinatorics for centraliser towers of finite-dimensional algebras arising from factors \(N\subset M\), \(M\) of type \(\text{II}_ 1\), and the classification of finite index subfactors of the hyperfinite \(\text{II}_ 1\) factor, or finite depth [see e.g., F. Goodman, P. de la Harpe and V. Jones, Coxeter graphs and towers of algebras (1989; Zbl 0698.46050)]. The related fields include braid groups on strings (which feature in quantum groups and solvable models of statistical mechanics, and thread through the entire article), Hecke algebras (complex algebras with generators related to the system of projection operators) and statistical mechanical models [cf. H. V. N. Temperley and E. H. Lieb, Proc. R. Soc. Lond., Ser. A 322, 251-280 (1971; Zbl 0211.567) who deal with tensor product representations of the projection systems]. Further developments dealt with the knot polynomials as introduced by the author in Bull. Am. Math. Soc., New Ser. 12, 103-111 (1985; Zbl 0564.57006) (New invariants for knots and links), positivity of the Markov trace (a special trace leading to \(\text{II}_ 1\) factors) on Hecke algebras [see H. Wenzl, Invent. Math. 92, No. 2, 349-383 (1988; Zbl 0663.46055)], the BMW algebra [J. Murakami, Osaka J. Math. 24, 745-758 (1987; Zbl 0666.57006), J. S. Birman and H. Wenzl, Trans. Am. Math. Soc. 313, No. 1, 249-273 (1989; Zbl 0684.57004)] and the connection of braid group representations of factors to conformal field theory. The author suggests a connection between braid group representations and algebraic quantum field theory (and in particular superselection sectors) and loop groups. The author gives some conjectures on the connection to loop groups via vertex operators [A. Tsuchiya and Y. Kanie, Adv. Stud. Pure Math. 16, 297-372 (1988; Zbl 0661.17021)] and to topological field theory. He also describes E. Wittens ideas on topological quantum field theory [Commun. Math. Phys. 121, No. 3, 351-399 (1989; Zbl 0667.57005)] and M. Atiyah [Publ. Math., IHES 68, 175-186 (1988; Zbl 0692.53053)].

Reviewer: A.Wulfsohn (Coventry)

### MSC:

46L10 | General theory of von Neumann algebras |

46L37 | Subfactors and their classification |

46N50 | Applications of functional analysis in quantum physics |

46L35 | Classifications of \(C^*\)-algebras |

46N55 | Applications of functional analysis in statistical physics |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |