Lattice of intermediate subalgebras

Abstract

Analogous to subfactor theory, employing Watatani's notions of index and C * -basic construction of certain inclusions of C * -algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital C * -algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate C * -subalgebras of any inclusion of unital C *algebras admitting a finite index conditional expectation. Then, on the lines of Bakshi et al. (Trans. Amer. Math. Soc. 371 (2019) 5973-5991), we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate C * -subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an irreducible inclusion of type III factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite. Contents