Factorization of operators on $C^*$-algebras

Abstract

Let $A$ be a $C^*$-algebra. It is shown that every absolutely summing operator from $A$ into $\ell_2$ factors through a Hilbert space operator that belongs to the 4-Schatten- von Neumann class. We also provide finite dimensinal examples that show that one can not improve the 4-Schatten-von Neumann class to $p$-Schatten von Neumann class for any $p<4$. As application, we prove that there exists a modulus of capacity $\epsilon \to N(\epsilon)$ so that if $A$ is a $C^*$-algebra and $T \in \Pi_1(A,\ell_2)$ with $\pi_1(T)\leq 1$, then for every $\epsilon >0$, the $\epsilon$-capacity of the image of the unit ball of $A$ under $T$ does not exceed $N(\epsilon)$. This aswers positively a question raised by Pe\l czynski.