The $$\lambda $$ λ -Function in the Space of Trace Class Operators

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 $$\begin{aligned} \lambda (a) = \frac{1 - \Vert a\Vert _1 + 2 \Vert a\Vert _{\infty }}{2}, \end{aligned}$$ for every a in $${C_1(H)}$$ with $$\Vert a\Vert _1 \le 1$$ . This is a non-commutative extension of the formula established by Aron and Lohman for $$\ell _1$$ .