In the Jones Index theory, Longo's sector theory has been a powerful approach to the analysis for inclusions of factors and canonical endomorphisms have played an important role. In this paper, two topics on commuting canonical endomorphisms are studied: For a composition of two irreducible inclusions of depth 2 factors, the commutativity of corresponding canonical endomorphisms is shown to be the condition for the ambient irreducible inclusion to be of depth 2, that is, to give a finite dimensional Kac algebra. And an equivalent relation between the commuting -commuting square condition and the existence of two simultaneous commuting canonical endomorphisms is discussed.
Takashi SANO. "On commuting canonical endomorphisms of subfactors." J. Math. Soc. Japan 55 (1) 131 - 142, January, 2003. https://doi.org/10.2969/jmsj/1196890846