The Birman–Schwinger principle in von Neumann algebras of finite type

We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman–Schwinger principle in this setting. As an application of this result, revisiting the Birman–Krein formula in the abstract scattering theory, we represent the de la Harpe–Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.