A simple question about positive element in C*-algebra I am reading a book about C*-algebra. There is a quotation below.
An $operator~system$ $E$ is a closed self-adjoint subspace of a unital C*-algebra $A$ such that $1_{A}\in E$. The $n \times n$ matrices over $E$, $M_{n}(E)$, inherit an order structure from $M_{n}(A)$: an element in $M_{n}(E)$ is positive if and only if it is positive in $M_{n}(A)$.
First, what is the definition of positive element in a operator system? Then, how to explain an element in $M_{n}(E)$ is positive if and only if it is positive in $M_{n}(A)$?
 A: You ask what is the definition of a positive element in $E$; the element $a \in E$ is positive if $a$ is positive when we think of it as an element in $A$. But an operator system is more than that. It also allows us to define an order structure on $M_n(E)$ for each $n$ and to say whether each matrix $[a_{ij}]\in M_n(E)$ is positive or not.
The statement "an element in $M_n(E)$ is positive if and only if it is positive in $M_n(A)$" is not a theorem that needs to be proved.; it is a definition. The order structure in $M_n(A)$ is used to define the order structure in $M_n(E)$. The matrix $[a_{ij}]\in M_n(E)$ is (by definition) positive if it is positive when we consider it as an element in $M_n(A)$.
So what is this matrix order on $M_n(A)$? If $A$ is a C$^*$-algebra, then $M_n(A)$ is also a C$^*$-algebra. If we think of the C$^*$-algebra $A$ as a subalgebra of $B(H)$, the bounded operators on a Hilbert space $H$, then $M_n(A)$ is a subalgebra of $B(H^n)$. (Here $H_n$ is the $\ell_2$-direct sum of $n$ copies of the Hilbert space $H$.) 
So the element $a=[a_{ij}]\in M_n(E)$ would be positive, if $a$ is a positive map from $H^n$ to $H^n$. That would mean that
$$
\sum_{i,j=1}^n (a_{ij} \xi_j \mid \xi_i ) \geq 0,
$$
for all vectors $\xi_1, \ldots, \xi_n \in H$.
We can also describe the order structure of $M_n(A)$ without fixing a Hilbert Space $H$. The following are Lemmas IV.3.1 and IV.3.2 from Takesaki's Theory of Operator Algebra, Volume 1.
Lemma IV.3.1. An element of $M_n(A)$ is positive if and only if it is a sum of matrices
of the form $[a_i^*a_j]$ with $a_1, \ldots, a_n \in A$.
Lemma IV.3.2. A matrix $a = [a_{ij}] \in M_n(A)$ is positive if and only if 
$$
\sum_{i,j=1}^n x_i^* a_{ij} x_j \geq 0,
$$
for every $x_1, \cdots x_n \in A$.
