A remark on non-commutative $L^p$-spaces

TL;DR

This paper provides explicit descriptions of non-commutative L^p-spaces for tensor product von Neumann algebras, linking them to individual components and demonstrating their usefulness in quantum information theory.

Contribution

It introduces explicit formulas for Haagerup and Kosaki non-commutative L^p-spaces of tensor product algebras, enhancing understanding and applications in quantum information.

Findings

Explicit descriptions of non-commutative L^p-spaces for tensor products.

Connections established between tensor product spaces and individual components.

Applications demonstrated in quantum information theory.

Abstract

We explicitly describe the Haagerup and the Kosaki non-commutative $L^p$-spaces associated with a tensor product von Neumann algebra $M_1\bar{\otimes}M_2$ in terms of those associated with $M_i$ and usual tensor products of unbounded operators. The descriptions are then shown to be useful in the quantum information theory based on operator algebras.