A note on irreducible quadrilaterals of II1 factors

Abstract

Given any quadruple [Formula: see text] of [Formula: see text]-factors with finite index, the notions of interior and exterior angles between [Formula: see text] and [Formula: see text] were introduced in [An angle between intermediate subfactors and its rigidity, Trans. Amer. Math. Soc. 371(8) (2019) 5973–5991]. We determine the possible values of these angles in terms of the cardinalities of the Weyl groups of the intermediate subfactors when [Formula: see text] is an irreducible quadrilateral and the subfactors [Formula: see text] and [Formula: see text] are both regular. For an arbitrary irreducible quadruple, an attempt is made to determine the values of angles by deriving expressions for the angles in terms of the common norm of two naturally arising auxiliary operators and the indices of the intermediate subfactors of the quadruple. Finally, certain bounds on angles between [Formula: see text] and [Formula: see text] are obtained when [Formula: see text] is regular, which enforce some restrictions on the index of [Formula: see text] in terms of that of [Formula: see text].