The projection constant for the trace class

TL;DR

This work identifies the projection constant of the trace class operators $\mathcal S_1(n)$ as an explicit Haar-average integral: $\boldsymbol{\lambda}(\mathcal{S}_1(n))=n\int_{\mathcal{U}_n}|\mathrm{tr}(V)|\,dV$. By embedding $\mathcal{S}_1(n)$ into the space of unitarily invariant functions on $\mathcal{U}_n$ via unitary harmonics, the authors apply Rudin’s averaging technique to obtain a sharp, convolution-based description of the relevant projection, and prove the integral formula and the limit $\lim_{n\to\infty}\boldsymbol{\lambda}(\mathcal{S}_1(n))/n=\sqrt{\pi}/2$ using Weingarten calculus and probabilistic limit results for $\mathrm{tr}(U)$. The approach unifies non-commutative projection-constant theory with harmonic analysis on compact groups and random-matrix techniques, yielding precise finite-dimensional formulas and asymptotics. This advances the understanding of local theory of Banach spaces in the non-commutative Schatten setting and provides explicit asymptotics with potential applications in operator algebras and quantum information.

Abstract

We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, $\mathcal S_1(n)$. We show an integral formula for the projection constant of $\mathcal S_1(n)$; namely $ \boldsymbolλ\big(\mathcal S_1(n)\big) = n \int_{\mathcal U_n} \vert \text{tr}(U) \vert \,dU \,, $ where the integration is with respect to the Haar probability measure on the group $\mathcal U_n$ of unitary operators. Using a probabilistic approach, we derive the limit formula $ \lim_{n\to \infty} \boldsymbolλ\big(\mathcal S_1(n)\big)/n = \sqrtπ/2\,. $

Theorems & Definitions (43)

• Proposition 1.1
• proof
• Lemma 1.2
• proof
• Proposition 1.3
• Lemma 1.4
• proof
• Lemma 1.5
• proof
• Lemma 1.6