On commuting squares and subfactors (Q1180530)

An explicit construction of irreducible selfdual subfactors of the hyperfinite factor \(R\) by using a certain type of commuting squares of matrix algebras are presented. Examples so constructed give some information about the set \(I^ 0_ R\) of index values of irreducible (selfdual) subfactors of \(R\) and its derived sets: \({\mathcal D}^ N(I^ 0_ R)=\) the set of all accumulation points of \({\mathcal D}^{N-1}(I^ 0_ R)\) \((N=1,2,\ldots)\). For example, for any positive integer \(N\), \((N+(N^ 2+4)^{1/2})/2\), \((N+(N^ 2+8)^{1/2})/2\in I^ 0_ R\), \((N+N^{-1})^ 2\in{\mathcal D}(I^ 0_ R)\), \((N+1)^ 2\in{\mathcal D}^ N(I^ 0_ R)\). The construction is based on the matrix \[ L= \left[\begin{matrix} 0 &1 &0 &\ldots && 0 \\ 1 &N_ 2 &1 &\ldots &&0 \\ 0 &1 &N_ 3 &\ldots &&0 \\ \vdots &\vdots &\vdots &&&\vdots \\ 0 &. &. &.1 &N_{n-1} &1 \\ 0 &. &. &.0 &1 &N_ n \end{matrix} \right], \] where \(N_ 2,N_ 3, \ldots, N_ n\) are non-decreasing sequences of non-negative integers. \(\| L\|^ 2\) gives the index.