Factorization of operators on C*-algebras

Let A be a C*-algebra. We prove that every absolutely summing operator from A into ${\displaystyle {\ell }_2}$ factors through a Hilbert space operator that belongs to the 4-Schatten-von Neumann class. We also provide finite-dimensional examples that show that one cannot replace the 4-Schatten-von Neumann class by the p-Schatten-von Neumann class for any p < 4. As an application, we show that there exists a modulus of capacity ε → N(ε) so that if A is a C*-algebra and ${\displaystyle T\in {\Pi }_1(A,{\ell }_2)}$ with ${\displaystyle {\pi }_1\left(T\right)\le 1}$, then for every ε >0, the ε-capacity of the image of the unit ball of A under T does not exceed N(ε). This answers positively a question raised by Pełczyński.