Let $A\subset B(H)$ be an operator system and $B$ be a $C^*$-algebra. (1)$u:A\rightarrow B$ is called a decomposable map if $u$ is in the linear span of $CP(A, B)$, where $CP(A,B)$ is the set of all completely positive maps from $A$ to $B$.
There is another equivalent definition. (2)$u:A\rightarrow B$ is called a decomposable map if $u=u_1-u_2+i(u_3-u_4)$, each $u_i\in CP(A,B)(i=1,\cdots 4) $
It is easy to see that (2) implies (1). How to check that (1) imply (2)?
Suppose $u:A\to B$ is decomposable, and write $u=\sum_{k=1}^n\lambda_ku_k$ for some $\lambda_k\in\mathbb C$ and $u_k\in CP(A,B)$. Each $\lambda_k$ can be written $\lambda_k=(\lambda_{k,1}-\lambda_{k,2})+i(\lambda_{k,3}-\lambda_{k,4})$, where $\lambda_{k,j}\geq0$ for $k=1,\ldots,n$, $j=1,\ldots,4$. Then for each $j$, $\sum_{k=1}^n\lambda_{k,j}u_k$ is in $CP(A,B)$, and we have $$u=\left(\sum_{k=1}^n\lambda_{k,1}u_k-\sum_{k=1}^n\lambda_{k,2}u_k\right)+i\left(\sum_{k=1}^n\lambda_{k,3}u_k-\sum_{k=1}^n\lambda_{k,4}u_k\right).$$