Factors generated by direct sums of \(II_1\) factors (Q1363222)

Let \(\{M_{n}\}_{n\in N}\) and \(\{N_{n}\}_{n\in N}\) be two increasing sequences of finite direct sums of \(II_{1}\) factors such that for every \(n\), the following diagram \[ \begin{matrix} M_{n} & \subset & M_{n+1}\\ \cup & &\cup\\ N_{n} & \subset & N_{n+1}\end{matrix} \] is a commuting square. Let \(M= (\cup_{n}M_{n})''\) and \(N=( \cup_{n}N_{n})''\). The author shows that if the inclusion relations \(N_{n}\subset N_{n+1}\), \(M_{n}\subset M_{n+1}\), and \(N_{n}\subset M_{n}\) are periodic, then \(M\) and \(N\) are \(II_{1}\) factors and provides a formula for the Jones index of \(N\subset M\).