Angles Between Intermediate Operator Subalgebras

Motivated by [2] and [5], the notions of interior and exterior angles between a pair of compatible intermediate W * -subalgebras of an inclusion of W * -algebras with a normal conditional expectation with finite probabilistic index are introduced. This is then employed effectively to define the interior angle between a pair of compatible intermediate C * -subalgebras of an inclusion of non-unital C * -algebras with a conditional expectation with finite Watatani index. It is also shown that the interior angle is stable under the minimal tensor product of unital C * -algebras.