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The complex Hadamard matrices of dimension $n$ are used to build index $n$ subfactors through the commuting square construction. For more details, see the paper Subfactors and Hadamard Matrices by W. Camp and R. Nicoara.

In his PhD dissertation (1997), M. Petrescu has built three continuous families of complex Hadamard matrices of dimension $7$, see these matrices in this link.

Question: Are there infinitely many isomorphism class of amenable subfactors coming from these (dimension $7$) Hadamard matrices of Petrescu?

Remark: A subfactor of index $7$ must be maximal (which is not necessarily true for the prime numbers greater than $8$). So if the question admits a positive answer, then the index $7$ would be the first known index allowing infinitely many non-isomorphic maximal amenable subfactors (see here).

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  • $\begingroup$ The assumption "amenable" is useful because in p14 of the paper A finiteness result for commuting squares of matrix algebras of R. Nicoara: << A main point of interest in this (Petrescu’s) result is that it can produce one-parameter families of non-isomorphic subfactors with (the same index $n$ and) the same graph, conjectured to be $A_∞$ >> $\endgroup$
    – Sebastien Palcoux
    Commented Apr 11, 2015 at 14:43

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