On commuting squares and subfactors

Abstract

After showing (see Theorem 2.8) how commuting squares lead, in the presence of a certain additional "rotational symmetry" condition, to irreducible subfactors, with pleasant additional features, of the hyperfinite II₁ factor R, it is shown (see Theorem 3.1) that such rotationally symmetric commuting squares can be constructed, starting from each member of a certain class of symmetric non-negative integral matrices. Specialisations of the matrix show (see Sect. 4), for instance, that for each positive integer N, (N+√N²+4)/2 and (N+√N²+8)/2 belong to the set 1_R⁰ of index-values of irreducible subfactors of R, that (N+1/N)² is an accumulation point of 1_R⁰, and that 9 is an accumulation point of accumulation points of 1_R⁰, 16 is an accumulation point of accumulation points of accumulation points of 1_R⁰, and so on.