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 I I I 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.