A representation of a $ C^{*} $-algebra.

Question

I have a quick question about the representation theory of $ C^{*} $-algebras.

A representation of a $ C^{*} $-algebra $ A $ is a $ * $-homomorphism $ \pi: A \to B(\mathcal{H}) $, where $ \mathcal{H} $ is a Hilbert space and $ B(\mathcal{H}) $ is the $ C^{*} $-algebra of all bounded linear operators on $ \mathcal{H} $.

Question. Does such a $ * $-homomorphism exist all the time? If so, how do we know that such a $ * $-homomorphism always exists?

Thanks!

Answer 1

The fact that such a $ * $-homomorphism exists is the statement of the Gelfand-Naimark-Segal (GNS) Construction.