On commutativity of diagrams of type \(\Pi_1\) factors (Q1127682)

Let \(\mathcal D\) denote a square diagram composed of type \(\text{II}_1\) factor inclusions \(\text{S}\subset Q\subset K\) and \(\text{S}\subset R\subset K\) with \(\text{Q}\not\subset R, R\not\subset Q\) and assume that \(\text{[K:S]}<\infty\) and \(\text{ S}'\cap K = \mathbb C\). After preliminaries on properties of Jones projections (and their Ocneanu convolutions) arising from \(\mathcal D\) and related diagrams, it is shown that \(\mathcal D\) is a commuting square, provided that the second relative commutant of the inclusion \(\text{S}\subset \text{Q}\) has dimension 2 and \(\text{[Q:S]}>[R:S]\). Otherwise, commutativity of \(\mathcal D\) follows when \(\text{[Q:S]} = 4\cos^{2}(\pi /n)\) for a prime n. These and further results are given, mainly, in equivalent terms of co-commutativity of related diagrams [cf. \textit{T. Sano} and \textit{Y. Watatani}, J. Oper. Theory 32, No. 2, 209-241 (1994; Zbl 0838.46052)].