Sharp tangent inequalities in noncommutative L_p-spaces

Abstract

Let ℳ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let L_p(ℳ) denote the associated Haagerup noncommutative L_p-space. For 1<p<∞, let q=p/(p−1). For z∈ L_q(ℳ) and y∈ L_p(ℳ), denote by ⟨ z,y⟩=ReTr(z^*y) the canonical real duality pairing between L_q(ℳ) and L_p(ℳ). Define the normalized duality map by J_p(x)=|x|_p^2−pu|x|^p−1 for x=u|x|≠0, and set J_p(0)=0. We establish the sharp tangent inequality |x+y|_p²≥|x|_p²+2⟨ J_p(x),y⟩+(p−1)|y|_p², x,y∈ L_p(ℳ), 1<p≤2. The reverse inequality holds for 2≤ p<∞, and the constant p−1 is optimal. Our inequality extends Xu's tangent inequality \cite{Xu89} from classical L_p-spaces to general noncommutative L_p-spaces and provides the sharp tangent formulation of the Ricard--Xu convexity inequality \cite{RX16}. When specialized to Schatten p-classes, it also yields the corresponding tangent formulation of the Ball--Carlen--Lieb convexity inequality \cite{BCL94}. Under a suitable positivity assumption, we further establish a sharp complement to Xu's tangent inequality for Schatten p-classes in the range 0<p<1.

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BibTeX RIS

BibTeX

@misc{xu2026sharp,
  title = {{Sharp tangent inequalities in noncommutative $L_p$-spaces}},
  author = {Zongben Xu and Teng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16202},
  eprint = {2609.16202},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16202},
}

RIS

TY  - UNPB
TI  - Sharp tangent inequalities in noncommutative L_p-spaces
AU  - Zongben Xu
AU  - Teng Zhang
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.16202
UR  - https://arxiv.org/abs/2609.16202
AB  - Let ℳ be a von Neumann algebra equipped with a faithful normal semifinite weight, and let L_p(ℳ) denote the associated Haagerup noncommutative L_p-space. For 1<p<∞, let q=p/(p−1). For z∈ L_q(ℳ) and y∈ L_p(ℳ), denote by ⟨ z,y⟩=ReTr(z^*y) the canonical real duality pairing between L_q(ℳ) and L_p(ℳ). Define the normalized duality map by J_p(x)=|x|_p^ 2−pu|x|^p−1 for x=u|x|≠0, and set J_p(0)=0. We establish the sharp tangent inequality |x+y|_p^2≥|x|_p^2+2⟨ J_p(x),y⟩+(p−1)|y|_p^2, x,y∈ L_p(ℳ), 1<p≤2. The reverse inequality holds for 2≤ p<∞, and the constant p−1 is optimal. Our inequality extends Xu's tangent inequality \cite{Xu89} from classical L_p-spaces to general noncommutative L_p-spaces and provides the sharp tangent formulation of the Ricard--Xu convexity inequality \cite{RX16}. When specialized to Schatten p-classes, it also yields the corresponding tangent formulation of the Ball--Carlen--Lieb convexity inequality \cite{BCL94}. Under a suitable positivity assumption, we further establish a sharp complement to Xu's tangent inequality for Schatten p-classes in the range 0<p<1.
ER  -

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