Von Neumann entropy of the angle operator between a pair of intermediate subalgebras

Given a pair of intermediate $C^{\ast }$-subalgebras of a unital inclusion of simple $C^{\ast }$-algebras with a conditional expectation of finite Watatani index, we discuss the corresponding angle operator and its Fourier transform. We provide a calculable formula for the von Neumann entropy of the (Fourier) dual angle operator for a large class of quadruple of simple $C^{\ast }$-algebras.