In N&Chuang, on page 368 is written the following theorem:
The map $\mathcal{E}$ satisfies axioms A1,A2,A3 if and only if $$\mathcal{E}(\rho)=\sum_k E_k \rho E_k^{\dagger}$$ Where $\sum_k E_k^{\dagger} E_k \leq I$
The axiom A2 is convex linearity, the axiom A3 is CP, the axiom A1 is:
Axiom A1: $0 \leq Tr(\mathcal{E}(\rho)) \leq 1$
Shouldn't be added in the theorem: $\sum_k E_k^{\dagger} E_k \geq 0$ as well to ensure the fact the trace can never be negative ? So in the end we would have:
$$0 \leq \sum_k E_k^{\dagger} E_k \leq I$$