What question did this study set out to answer?

The aim is to establish generalized Cauchy–Schwarz inequalities for positive sesquilinear maps in noncommutative L^p spaces, yielding applications such as uncertainty relations.

May 17, 2026Open Access

Cauchy–Schwarz Inequalities for Maps in Noncommutative Lᵖ-Spaces

Key Points

The aim is to establish generalized Cauchy–Schwarz inequalities for positive sesquilinear maps in noncommutative L^p spaces, yielding applications such as uncertainty relations.
Developed generalized Cauchy–Schwarz inequalities for positive sesquilinear maps in noncommutative L^p spaces for p>1.
Provided bound estimates for real and imaginary parts of these maps.
Proposed a new norm on noncommutative L^2 spaces, generalizing the numerical radius norm of bounded linear operators.
Established a Cauchy–Schwarz inequality for positive sesquilinear maps into noncommutative L^2 spaces with the new norm.
Derived a generalization of the uncertainty relation within noncommutative L^2 spaces.
Presented representations of general positive linear maps into noncommutative L^p spaces and other operator spaces.

Abstract

Abstract In this paper, some generalized Cauchy–Schwarz inequalities for positive sesquilinear maps with values in noncommutative Lᵖ L p -spaces for p>1 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 L 2 -spaces is given. Next, a new norm on a noncommutative L² L 2 -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 L 2 -space equipped with this new norm is proved. These results are used to get representations of general positive linear maps with values into a noncommutative Lᵖ L p -space and into certain operator spaces in several different situations. Some concrete examples are also given.

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