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Table 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.