How Does Positivity in C*-Algebras Relate to Their Representations?

[Oxymoron]

If I have a faithful nondegenerate representation of a C*-algebra, A:

$$\pi\,:\,A \rightarrow B(\mathcal{H})$$

where $B(\mathcal{H})$ is the set of all bounded linear operators on a Hilbert space. And just suppose that $a\geq 0 \in A$. How is the fact that a is positive got anything to do with $\pi(a)$ being positive?

Apparantly there is an if and only if relationship!? How does one begin to prove something like $a\geq 0 \Leftrightarrow \pi(a) \geq 0 \in B(\mathcal{H})$?

 

[matt grime]

Define positive for me again for a C* alg.

 

[Oxymoron]

Let $A$ be a C*-algebra, and $a,b \in A$. Then $a\geq b$ if $a-b$ has the form $c^*c$ for some $c\in A$. In particular, $a$ is positive if $a = c^*c$ for some $c \in A$.

 

[matt grime]

Well, one implication is obvious a positive implies pi(a) positive since pi is a *-homomorphism.

Conversely, hmm, well, faithfulness and nondegeneracy must come into it somewhere.