Lattice of intermediate subalgebras

Analogous to subfactor theory, employing Watatani's notions of index and $C^*$-basic construction certain inclusions $C^*$-algebras, (a) we develop a Fourier theory (consisting transforms, rotation maps shift operators) on the relative commutants any inclusion simple unital $C^*$-algebras with finite Watatani index, (b) introduce interior exterior angles between intermediate $C^*$-subalgebras admitting conditional expectation. Then, lines [2], apply these concepts obtain bound for cardinality lattice irreducible as in (a), improve Longo's subfactors an type $III$ factors index. Moreover, also show that fairly large class von Neumann algebras, subalgebras is always finite.