Cauchy–Schwarz Inequalities for Maps in Noncommutative $$L^p$$-Spaces

In this paper, some generalized Cauchy–Schwarz inequalities for positive sesquilinear maps with values in noncommutative L^p-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 L^2-spaces is given. Next, a new norm on a noncommutative 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 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 Lìp-space and into certain operator spaces in several different situations. Some concrete examples are also given.