The $\lambda$-function in the space of trace class operators

TL;DR

This paper proves that the space of trace class operators on a Hilbert space satisfies the $\lambda$-property and explicitly determines the $\lambda$-function, extending a known formula from commutative to non-commutative settings.

Contribution

It establishes the $\lambda$-property for trace class operators and derives an explicit formula for the $\lambda$-function in this non-commutative context.

Findings

Proves $C_1(H)$ satisfies the $\lambda$-property.

Determines the explicit form of the $\lambda$-function for trace class operators.

Extends the $\lambda$-function formula from $\ell_1$ to non-commutative operator spaces.

Abstract

Let $C_1(H)$ denote the space of all trace class operators on an arbitrary complex Hilbert space $H$. We prove that $C_1(H)$ satisfies the $\lambda$-property, and we determine the form of the $\lambda$-function of Aron and Lohman on the closed unit ball of $C_1(H)$ by showing that $$\lambda (a) = \frac{1 - \|a\|_1 + 2 \|a\|_{\infty}}{2},$$ for every $a$ in ${C_1(H)}$ with $\|a\|_1 \leq 1$. This is a non-commutative extension of the formula established by Aron and Lohman for $\ell_1$.