Asymptotic Spectral Measures, Quantum Mechanics, and E-theory

Asymptotic Spectral Measures: Between Quantum Theory and E-theory

We review the relationship between positive operator-valued measures (POVMs) in quantum measurement theory and asymptotic morphisms in the C*-algebra E-theory of Connes and Higson. The theory of asymptotic spectral measures, as introduced by Martinez and Trout (CMP 226), is integrally related to positive asymptotic morphisms on locally compact spaces via an asymptotic Riesz Representation Theorem. Examples and applications to quantum physics, including quantum noise models, semiclassical limits, pure spin one-half systems and quantum information processing will also be discussed.

通过渐近态射和准投影的奇谱局部化

The odd spectral localiser via asymptotic morphisms and quasi-projections

本文通过渐近态射和准投影研究奇K理论类与奇无界Kasparov模的索引配对，提出奇谱局部化的实例。

Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds

We consider the spectral behavior and noncommutative geometry of commutators $[P,f]$, where $P$ is an operator of order $0$ with geometric origin and $f$ a multiplication operator by a function. When $f$ is Hölder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudo-differential calculus is available, variations of Connes' residue trace theorem and related integral fo…