  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 [2], 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 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.
