Periodic commuting squares of finite von Neumann algebras (Q1384477)

The von Neumann algebras for the inclusions, say \(N \subset M\), are unless otherwise indicated taken to be finite direct sums of finite factors with the same identity. The trace on \(N\) is assumed to be the restriction of the trace on \(M\) and the minimal central projections are denoted by \((q_{j})\) and \((p_i)\) respectively. The inclusion is called connected if the intersections of the centres \({\mathcal{Z}}(M) \cap {\mathcal{Z}}(N) \simeq {\mathbb C}\). The index matrix \(\Lambda^{M}_{N}\) and the trace matrix \(T^{M}_{N}\) are represented by \(\lambda_{i,j} = [M_{p_iq_j}:N_{p_iq_j}]^{ {1 \over 2}}\) if \(p_{i}q_{j} \neq 0\) and otherwise zero, and by \(c_{i,j} = tr_{p_{j}M}(p_{i}q_{j})\), respectively, where \(tr_{p_{j}M}\) is the trace on the factor \(p_{j}M\). Suppose \[ \begin{matrix} A_0 &\subset &B_0\\ \bigcap &&\bigcap\\ A_1 &\subset &B_1\end{matrix} \] is a commuting square, i.e., conditional expectations satisfy \(E_{A_0}^{A_{1}} E^{B_{1}}_{A_{1}} = E^{B_{0}}_{A_{0}} E^{B_{}}_{B_{0}}\), assumed in this article to be with respect to a common Markov trace for \(B_0 \subset B_{1}\) and \(A_1 \subset B_{1}\); all inclusions are assumed to be of finite index. The author's hypotheses for periodic commuting squares rely on Section 4.3 of \textit{F. M. Goodman, P. de la Harpe} and \textit{V. F. R. Jones} [``Coxeter graphs and towers of algebras'' (1989; Zbl 0698.46050)], henceforth referred to as GHJ. He has however changed the hypotheses to start off with large enough sums of factors, in that \(n_{0}\) of GHJ is assumed to be 0, which is acceptable, but he has not attempted to justify his assumption that the index and trace matrices are square, implicit in his use of matrix primitivity, i.e., aperiodicity in Chapter 1 of GHJ. Indeed these matrices are not generally square. However, it can be shown from the GHJ-axioms (say for 2-periodicity seeing that the author is concentrating on 2-periodicity) since the inclusion matrix \(\Lambda_n\) for \(A_{n} \subset A_{n+1}\) is, by the definition of periodicity, the same as \(\Lambda_{n+2}\) for \(A_{n+2} \subset A_{n+3}\), either of the square matrices \(\Lambda_{n+1} \Lambda_{n}\) or \(\Lambda_{n} \Lambda_{n+1}\) could be be square inclusion matrices for \(A_{n} \subset A_{n+2}\). Thus there is, up to pseudo-equivalence (see GHJ), one square inclusion matrix. It follows also from GHJ that from \(n_0\) onwards one carries on with the same square inclusion matrix at each step. Iterating basic constructions [see \textit{V. F. R. Jones}, Invent. Math. 72, 1-25 (1983; Zbl 0508.46040)] one gets projections \(e_{n} = e_{B_{n-1}}\) and finite von Neumann algebras \(B_{n+1} = \langle B_{n},e_{n}\rangle\) and one lets \(A_{n+1} = (A_{n} \cup \{ e_{n} \})\). Assuming all inclusions are connected, one may reverse these operations to get, for instance, \(A_{n+1}\) isomorphic to \(\langle A_{n}, e_{n}\rangle\). Then the \[ \begin{matrix} A_n &\subset &B_n\\ \bigcap &&\bigcap \\ A_{n+1} &\subset &B_{n+1} \end{matrix} \] are also commuting squares. Commuting squares are called 2-periodic if the \( T^{A_{n+1}}_{A_{n}}\) and \( T^{B_{n+1}}_{B_{n}}\) are periodic, and the \( T^{A_{n+2}}_{A_{n}}\) and \( T^{B_{n+2}}_{B_{n}}\) are aperiodic. The author proves that if all inclusions are connected, and if there is a \(\lambda\) such \(F^{A_{1}}_{A_{0}} = \lambda I_{n}\) and \( F^{B_{1}}_{B_{0}} = \lambda I_{m}\) where \(n = \dim_{\mathbb C}{\mathcal{ Z}}(A_{0})\), \(m = \dim_{\mathbb C}{\mathcal{ Z}}(B_{0})\), then the commuting squares are 2-periodic. For example, \(F^{A_{1}}_{A_{0}}\) is taken to be the diagonal matrix with entries \((\sum_{i=1}^{n}{\lambda^{2}_{i,j} \over c_{i,j}})^{-1}\). The author also proves, all inclusions being connected, that the commuting squares are symmetric under a right-angled rotation and that the towers of squares are transitive. He gives examples of periodic commuting squares where \(A_{0},B_{0}\) and \(B_{1}\) are \(II_{1}\)-factors but \(A_{1}\) is not a factor, for instance when \(B_{0}\) and \(B_{1}\) involve crossed products of \(A_{0}\) with a finite Abelian group.