Weak* decomposition and Radon-Nikodym theorem for quantum expectations

TL;DR

This paper develops a weak* decomposition theory for quantum expectations on non-commutative probability spaces, linking it to the Radon-Nikodym theorem and extending classical measure theory concepts to quantum settings.

Contribution

It introduces a novel weak* decomposition framework for quantum expectations and establishes its equivalence with the AGKL decomposition under the KMS condition.

Findings

Defined weak* continuity and singularity for quantum expectations.

Derived the weak* decomposition and Radon-Nikodym derivatives in quantum contexts.

Connected the decomposition to classical measure theory via the KMS condition.

Abstract

A quantum expectation is a positive linear functional of norm one on a non-commutative probability space (i.e., a C*-algebra). For a given pair of quantum expectations $\mu$ and $\lambda$ on a non-commutative probability space $A$, we propose a definition for weak* continuity and weak* singularity of $\mu$ with respect to $\lambda$. Then, using the theory of von Neumann algebras, we obtain the natural weak* continuous and weak* singular parts of $\mu$ with respect to $\lambda$. If $\lambda$ satisfies a weak tracial property known as the KMS condition, we show that our weak* decomposition coincides with the Arveson-Gheondea-Kavruk Lebesgue (AGKL) decomposition. This equivalence allows us to compute the Radon-Nikodym derivative of $\mu$ with respect to $\lambda$. We also discuss the possibility of extending our results to the positive linear functionals defined on the Cuntz-Toeplitz operator system.