Factorization of Operators on C

Let A be a C∗-algebra. We prove that every absolutely summing operator from A into l2 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 can not replace the 4-Schatten-von Neumann class by 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 T ∈ Π1(A, l2) with π1(T ) ≤ 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 Pe lczyński.