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.