Communications in Mathematical Physics - Volume 226 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Edu Received: 27 December 2000 / Accepted: 8 October 2001 Abstract: We study the relationship between POV-measures in quantum theory and asymptotic morphisms in the operator algebra E-theory of Connes–Higson. This is done by introducing the theory of “asymptotic” PV-measures and their integral correspondence with positive asymptotic morphisms on locally compact spaces. Examples and applications involving various aspects of quantum physics, including quantum noise models, semiclassical limits, strong deformation quantizations, and pure half-spin particles, are also discussed.

Also, the (non-closed) ideal Cc (X) of compactly supported functions is a (non-closed) hereditary ∗-subalgebra of Cb (X). However, in general, Cδ (X), for δ = c, 0, b, is not a hereditary subalgebra of the C ∗ -algebra Bb (X) of bounded Borel functions on X. 3. Asymptotic Spectral Measures In this section we give the basic definitions and properties of asymptotic spectral measures. See the Appendix for a review of POV and spectral measures. Let (X, ) be a measurable space and H a separable Hilbert space.

1. For each f ∈ L2 (C, dλ) ∩ Bb (C) the operator Tf ∈ K(F). Proof. 2 [G1]. We define the Wick quantization map QW : Cb (C) → B(F) : f → QW (f ) = Tf . Let P : H → F denote the orthogonal projection. We can then define a POV-measure AW : C → B(F) by AW ( ) = P ◦ χ , where χ denotes (the operator on H of multiplication by) the characteristic function χ . Note that it is the compression of the PVM → E( ) = χ . 2. The POVM AW : ( C , CC ) → (B(F), K(F)) is normalized and locally compact-valued. The associated positive linear map is the Wick-Toeplitz quantization QW : (Cb (C), C0 (C)) → (B(F), K(F)).

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