Free Rosenthal Inequalities on Vector-Valued Symmetric Operator Spaces

Abstract

Abstract Let $\mathcal{M}$ be a finite von Neumann algebra and let $E$ be an interpolation space between $L_1(0,1)$ and $L_{\infty }(0,1)$. For every $q\in [1,\infty ]$, we introduce a general notion of a vector-valued symmetric operator space $E(\mathcal{M};\ell _q)$ via $K$-functional of the couple $(L_1(\mathcal M;\ell _q),L_{\infty }(\mathcal M;\ell _q))$. We then establish free Rosenthal inequalities on $E(\mathcal{M};\ell _q)$, which provide an equivalent characterization of the quantity $\|(x_k)_{k\geq 0}\|_{E(\mathcal{M};\ell _q)},$ where $(x_k)_{k\geq 0}\subset E(\mathcal{M})$ is a sequence of freely independent random variables.