L$^p$ spaces of operator-valued functions

TL;DR

This paper introduces a new L^p space for operator-valued functions in quantum probability, establishing its completeness and exploring related inequalities, thus extending classical analysis to quantum operator contexts.

Contribution

It defines a novel p-norm for quantum random variables, creating a complete operator-valued L^p space and generalizing classical inequalities like Hölder's.

Findings

The operator-valued L^p space is complete.

Various norm candidates are analyzed.

Hölder's inequality is generalized to this setting.

Abstract

We define a p-norm in the context of quantum random variables, measurable operator-valued functions with respect to a positive operator-valued measure. This norm leads to a operator-valued L^p space that is shown to be complete. Various other norm candidates are considered as well as generalizations of H\"older's inequality to this new context.