Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces
Giorgia Bellomonte, Stefan Ivkovic, Camillo Trapani
TL;DR
The paper addresses the problem of extending Cauchy–Schwarz-type inequalities to positive sesquilinear maps valued in noncommutative $L^p$-spaces for $p>1$ and to operator-valued settings with numerical-radius norms. The authors develop a generalized Cauchy–Schwarz inequality with a factor $2$ for $L^p(\rho)$-valued maps, establish sharper bounds when the image is normal, and derive bounds for real and imaginary parts in the $L^2(\rho)$ setting, which yield uncertainty-relations-type results. They extend these results to ordered Banach bimodules and introduce a new numerical-radius-type norm on $L^2(\rho)$, proving CS inequalities in this framework and for associated operator-valued maps. A key contribution is a representation program: using these CS inequalities to realize positive maps as (quasi) Banach-space representations via generalized GNS constructions, with applications to quantum physics and operator theory. The practical impact lies in providing robust inequalities and norm-structuring tools that enable representations and analysis of positive maps into noncommutative $L^p$-spaces and related operator spaces, with concrete examples and links to uncertainty relations.
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.