A few remarks on intermediate subalgebras of an inclusion of C*-algebras

We show that the angle between intermediate $C^{\ast }$-subalgebras of an inclusion of simple $C^{\ast }$-algebras with finite Watatani index is stable. The notion of angle is instrumental in providing a bound for the cardinality of the lattice of intermediate subalgebras for an irreducible inclusion of simple $C^{\ast }$-algebras. We improve the existing upper bound for the cardinality of this set.