Why is it that for a unital $C^*$-algebra any positive linear functional $\varphi$ must be such that $\|\varphi\| =\varphi()$?

Question

I understand that this is a consequence of a related, more general theorem for non-unital $C^*$-algebras, where we can find approximants of a unit, but for the simple case of a unital $C^*$-algebra $\mathscr{A}$ , what is the simple argument that implies that positive linear functionals $\varphi:\mathscr{A} \rightarrow \mathbb{C}$ (defined as $\varphi(a^*a) \geq 0, \forall \, a \in \mathscr{A}$) are such that $\|\varphi\| =\varphi(\mathbf 1)$ ?

Answer 1

First $\varphi(\mathbf 1) \le \left\|\varphi\right\| \left\|\mathbf 1\right\| = \left\|\varphi\right\|$. For the reverse, we start with the following claim:

If $a\in\mathcal A$ is self-adjoint then $$\varphi (a) \in \mathbb R\quad \text{and} \quad \left|\varphi(a)\right| \le \left\|a\right\| \varphi\left(\mathbf 1\right)$$

Proof: Since $\left\|a\right\|\mathbf 1 \succeq a \succeq - \left\|a\right\|\mathbf 1$ then by linearity and positivity of $\varphi$, $$\left\|a\right\|\varphi\left(\mathbf 1\right) \ge \varphi(a) \ge -\left\|a\right\|\varphi(\mathbf 1)$$

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Now we are back to the general case, let $a\in \mathcal A$ (not mandatory self-adjoint),

\begin{align} \left|\varphi\left(a\right)\right|^2 &= \left|\varphi\left(\mathbf 1^* a\right)\right|^2\\ &\le \varphi(\mathbf 1)\left|\varphi\left(a^*a\right)\right| &&(\text{Cauchy-Schwarz})\\ &\le \varphi\left(\mathbf 1\right)\left\|a^* a\right\|\varphi\left(\mathbf 1\right) &&(\text{Apply the claim on $a^*a$})\\ &= \varphi(\mathbf 1)^2\left\|a\right\|^2 &&(\text{using $\left\|a^*a\right\| = \left\|a\right\|^2$)} \end{align}

This proves that $\left\|\varphi\right\| \le \varphi(\mathbf 1)$.