Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces

TL;DR

This paper extends the Cauchy-Schwarz inequality to positive sesquilinear maps in noncommutative Lp-spaces, providing bounds, a generalized uncertainty principle, and new norms, with applications to operator representations and concrete examples.

Contribution

It introduces new inequalities, bounds, and norms for positive sesquilinear maps in noncommutative Lp-spaces, generalizing classical results and norms.

Findings

Generalized Cauchy-Schwarz inequalities for noncommutative Lp-spaces.

Bound estimates for real and imaginary parts of sesquilinear maps.

Representation results for positive linear maps in noncommutative operator spaces.

Abstract

In this paper, a generalized Cauchy-Schwarz inequality for positive sesquilinear maps with values in noncommutative Lp-spaces for p > 1 are obtained. Bound estimates for their real and imaginary parts are also provided, and, as an application, a generalization of the uncertainty relation in the context of noncommutative L2-spaces are given. Next, a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von Neumann algebra into a C*-algebra equipped with the numerical radius norm is proved. In the same spirit, a new norm on a noncommutative L2-space, which generalizes the classical numerical radius norm of bounded linear operators on a Hilbert space, is proposed, and a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von-Neumann algebra into the noncommutative L2-space equipped with this new norm is proved. These results are used to get representations of general positive linear maps with values into a non-commutative Lp-space and into certain operator spaces in several different situations. Some concrete examples are also given.