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 a...