Irreducible and faithful representations of C*-algebra

Question

I am having some confusions regarding the direct sum of GNS representations of a C*-algebra $\mathcal{A}$. I know the GNS representations $(\mathcal{H}_f, \pi_f, x_f)$, corresponding to pure states $f$, are irreducible, meaning there's no closed subspace of $\mathcal{H}_f$ that is invariant under the action of $\mathcal{A}$. But does that give us surjectivity of $\pi_f$? Does $\mathcal{A}$ being unital ensure that?

Also, I'm aware the direct sum $\oplus_{f}(\mathcal{H}_f, \pi_f, x_f)$ of all the GNS representations corresponding to pure states $f$ is faithful, meaning it is injective. But does that mean it is isomorphic?

The confusion is when I try to deduce an element $a\in\mathcal{A}$ is positive when $\oplus_f \pi_f(a)$ is a positive definite operator in $\mathcal{B}\left(\oplus_f \mathcal{H}_f\right)$. Just because the representation is faithful, why do we know $a$ is also positive?

Also, if we know $f(a)>0$ for every pure state $f$, can we easily conclude $\oplus_f \pi_f(a)$ is an invertible operator? If it is, again, how can we deduce by faithfulness that $a$ is invertible in $\mathcal{A}$?

This is related to this and this questions.

Answer 1

The direct sum $$\bigoplus_{f\in P(A)}(\mathcal{H}_f,\pi_f, x_f): A \to B\left(\bigoplus_{f \in P(A)}\mathcal{H}_f\right)$$is indeed faithful. You ask: "Is it isomorphic?". I'm not sure what you mean to ask here, but it is definitely not an isomorphism of $C^*$-algebras (it is not surjective: your image only contains "diagonal" operators and does not see any operator on which the building blocks of the direct sum interplay).

The following is a general result:

If $\pi: A \to B$ is a faithful $*$-morphism, then $a$ is positive in $A$ if and only if $\pi(a)$ is positive in $B$.

Proof: Corestriction gives a $*$-isomorphism $A\to \pi(A)$ and thus $a$ is positive in $A$ if and only if $\pi(a)$ is positive in $\pi(A)$. However, $\pi(a)$ is positive in $\pi(A)$ if and only if $\pi(a)$ is positive in $B$ (one direction is a non-trivial, but well-known result).

Thus $a$ is positive if and only if $\bigoplus_{f\in P(A)}\pi_f(a)$ is a positive operator.

I don't see how to finish your question using this approach.