The spectrum of spin model angle operators

Abstract

Complex Hadamard matrices are biunitaries for spin model commuting squares. The corresponding subfactor standard invariant can be identified with the $1$-eigenspace of the angle operator defined by Jones. We identify the angle operator as an element of the symmetric enveloping algebra and compute its trace. We then show the angle operator spectrum coincides with the principal graph spectrum up to a constant iff the subfactor is amenable. We use this to show Paley type $II$ Hadamard matrices and Petrescu's $7 \times 7$ family of complex Hadamard matrices yield infinite depth subfactors.