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A recent paper on subfactors of von Neumann factors has stimulated much research in von Neumann algebras. It was discovered soon after the appearance of this paper that certain algebras which are used there for the analysis of subfactors could also be used to define a new polynomial invariant for links. Recent efforts to understand the fundamental nature of the new link invariants has led to connections with invariant theory, statistical mechanics and quantum theory. In turn, the link invariants, the notion of a quantum group, and the quantum Yang-Baxter equation have had a great impact on the study of subfactors. Our subject is certain algebraic and von Neumann algebraic topics closely related to the original paper. However, in order to promote, in a modest way, the contact between diverse fields of mathematics, we have tried to make this work accessible to the broadest audience. Consequently, this book contains much elementary expository material.
List of contents
1. Matrices over the natural numbers: values of the norm, classification, and variations.- 1.1. Introduction.- 1.2. Proof of Kronecker's theorem.- 1.3. Decomposability and pseudo-equivalence.- 1.4. Graphs with norms no larger than 2.- 1.5. The set E of norms of graphs and integral matrices.- 2. Towers of multi-matrix algebras.- 2.1. Introduction.- 2.2. Commutant and bicommutant.- 2.3. Inclusion matrix and Bratteli diagram for inclusions of multi-matrix algebras.- 2.4. The fundamental construction and towers for multi-matrix algebras.- 2.5. Traces.- 2.6. Conditional expectations.- 2.7. Markov traces on pairs of multi-matrix algebras.- 2.8. The algebras A?,k for generic ?.- 2.9. An approach to the non-generic case.- 2.10. A digression on Hecke algebras.- 2.11. The relationship between A?,n and the Hecke algebras.- 3. Finite von Neumann algebras with finite dimensional centers.- 3.1. Introduction.- 3.2. The coupling constant: definition.- 3.3. The coupling constant: examples.- 3.4. Indexfor subfactors of II1 factors.- 3.5. Inclusions of finite von Neumann algebras with finite dimensional centers.- 3.6. The fundamental construction.- 3.7. Markov traces on EndN(M), a generalization of index.- 4. Commuting squares, subfactors, and the derived tower.- 4.1. Introduction.- 4.2. Commuting squares.- 4.3. Wenzl's index formula.- 4.4. Examples of irreducible pairs of factors of index less than 4, and a lemma of C. Skau.- 4.5. More examples of irreducible paris of factors, and the index value 3 + 31/2.- 4.6. The derived tower and the Coxeter invariant.- 4.7. Examples of derived towers.- Appendix I. Classification of Coxeter graphs with spectral radius just beyond the Kronecker range.- I.1. The results.- I.2. Computations of characteristic polynomials for ordinary graphs.- I.3. Proofs of theorems I.1.2 and I.1.3.- Appendix II.a. Complex semisimple algebras and finite dimensional C*-algebras.- Appendix III. Hecke groups and other subgroups of PSL(2,?) generated by parabolic pairs.- References.
