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

Abstract

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 HarpeSkandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.

An integer-valued version of the Birman—Krein formula

“…[2,1,11,15,29,28,14,13,8,4,19,16,17]); its properties were reviewed and proven in a systematic fashion in [25]. For λ < inf σ ess (A), both projections E A (λ), E B (λ) have finite rank and so by (2.8) we have…”

On Properties of the ξ-Function in Semi-finite von Neumann Algebras

“…Similarly to the case of a finite A [25], we define the ξ-index via the Ξ-operators. We recall that the Ξ-operator Ξ(A) associated with an operator A in D A equals 1 π Im log A, where the operator logarithm is provided by the Dunford-Riesz functional calculus (cf.…”

“…We recall that the Ξ-operator Ξ(A) associated with an operator A in D A equals 1 π Im log A, where the operator logarithm is provided by the Dunford-Riesz functional calculus (cf. [20,25]). Whenever A is self-adjoint, Ξ(A) simplifies to the spectral projection E A (R − ).…”

“…Here det τ (S(λ)) is the de la Harpe-Skandalis determinant [23] of S(λ) (cf. [25]) and det(S(λ)) is the Fredholm determinant of S(λ). Formulae (4.6) and (4.7) were obtained in [25] and [21], respectively.…”

“…In its turn, this discovery helped to extend the range of situations for which the ξ-function can be defined [2]. Another prominent example is that an analog of the Birman-Schwinger principle for dissipative operators (that is, operators with non-negative imaginary part) in a finite von Neumann algebra obtained in [25] plays a principal role in the proof of a finite von Neumann algebra version of the Birman-Krein formula.…”