Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces

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.

💡 Research Summary

The paper investigates Cauchy‑Schwarz type inequalities for positive sesquilinear maps whose values lie in non‑commutative Lp‑spaces (Lp(ρ) with p > 1) and in operator spaces equipped with the numerical‑radius norm. The authors first recall the necessary background on von Neumann algebras M equipped with a faithful normal semifinite trace ρ, the associated non‑commutative Lp‑spaces, and the duality ⟨A,B⟩ = ρ(AB). A sesquilinear map Φ : X × X → Y (Y = Lp(ρ) or a space of bounded operators) is called positive if Φ(x,x) ≥ 0 for all x, and Hermitian when Φ(y,x)=Φ(x,y)∗.

Main results in Lp‑spaces.
Proposition 3.1 shows that for any positive sesquilinear Φ with values in Lp(ρ) (1 < p < ∞) one has
 ‖Φ(x,y)‖p ≤ 2‖Φ(x,x)‖p½‖Φ(y,y)‖p½ for all x,y∈X.
The factor 2 appears because the usual proof of the Cauchy‑Schwarz inequality fails when p > 1; the authors replace it by a decomposition of Φ(x,y) into a sum of two positive terms and apply Hölder’s inequality together with the trace’s normality.

Proposition 3.3 gives separate bounds for the real and imaginary parts:
 ‖Re Φ(x,y)‖p,‖Im Φ(x,y)‖p ≤ ‖Φ(x,x)‖p½‖Φ(y,y)‖p½.
These estimates are crucial for later applications to uncertainty relations.

When Φ(x,y) is a normal operator, the factor 2 can be removed. Proposition 3.7 proves that under the normality assumption the proper Cauchy‑Schwarz inequality holds:
 ‖Φ(x,y)‖p ≤ ‖Φ(x,x)‖p½‖Φ(y,y)‖p½.
The proof relies on the Kadison‑Schwarz inequality for normal elements and on the fact that normal operators commute with their adjoints, allowing a sharper estimate.

Using these bounds, the authors derive a non‑commutative version of the Heisenberg uncertainty principle (Proposition 3.4). For self‑adjoint A,B∈M, setting Φ(A,B)=ρ(AB) yields
 ‖Φ(A,A)‖2 ‖Φ(B,B)‖2 ≥ ¼|ρ(