The Knuth-Bendix algorithm is a practical algorithm which, for a given finite presentation of a group, finds a finite confluent set of relations, if it terminates. From this confluent set we can know a solution of the word problem for the group. In this paper we introduce a concept of a regular confluent set of relations, which also gives a solution of the word problem, and represent a class of groups with such sets in terms of the Cayley graphs of the groups. Furthermore we make an algorithm to find a regular confluent set of relations as an improvement of the Knuth-Bendix algorithm. This new algorithm is a genuine improvement because there are some finite presentation, for which the Knuth-Bendix algorithm does not stop but the new one does.

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We investigate the tensor product \mathscr{T}=V(λ¹)⊗…⊗V(λ^m) of the finite dimensional irreducible \mathscr{G}=gl(r, C) modules labelled by partitions λ¹, …, λ^m of m not necessarily distinct numbers n₁, …, n_m respectively. We determine the centralizer algebra End_\mathscr{G}(\mathscr{T}) and the projection maps of \mathscr{T} onto its irreducible \mathscr{G}-summands and give an explicit construction of the corresponding maximal vectors. In the special case that n_i=1 for i=1, …, m, the results reduce to the well-known results of Schur and Weyl.

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We consider an inclusion of type III factors with index 4 arising from an automorphism. It has the extended Coxeter-Dynkin diagram ˜{A} or A_{∞, ∞} as the principal graph. We show the equivalence between classification of automorphisms and that of the corresponding subfactors, and compute conjugacy invariants for the subfactors. As an application, we show the existence of a pair of type III₁ factors which does not split into a type II₁ inclusion.

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We study the operator relation ab=qba, where |q|=1, for self-adjoint operators.

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Two subfactors constructed from two factor maps are considered. A characterization is given for the angles between two subfactors to become trivial.

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Algebraic features of modular theory in von Neumann algebras are discussed with the help of Haagerup's L^p-theory and some notational devices.

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