Definition
Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.)
Problem
Given a C*-algebra $1\notin\mathcal{A}$ and adjoin a unit $\mathcal{A}\oplus\mathbb{C}$.
Consider a linear functional $\omega:\mathcal{A}\to\mathbb{C}$.
Then it has a canonical extenion: $$\omega_E(X+x1):=\omega(X)+x\|\omega\|$$
Then does it remain bounded and positive: $$\omega\geq0\implies\omega_E\geq0$$ $$\|\omega\|<\infty\implies\|\omega_E\|<\infty$$ Especially, does it maintain norm: $\|\omega_E\|=\|\omega\|$
(Revisiting Hahn-Banach, this extension may be even discontinuous!)
The problem is that I don't get ahead at: $$X+x1\geq0:\quad\omega_E(X+x1)=\omega(Y^2)+2y\omega(Y)+y^2=\ldots$$ $$\|X+x1\|<1:\quad|\omega_E(X+x1)|\leq|\omega(X)|+|x|=\ldots$$ Exploiting the representation: $X+x1=(Y+y1)(Y+y1):\quad Y=Y^*,y=\overline{y}$
Reference
I found a treatment of this in: Bratelli & Robinson
I finally figured it out; both separately...
Suppose it holds: $\omega(A^*A)\geq0$
Positivity can be proven by Kadison's inequality: $$\omega_E\left((A+a1)^*(A+a1)\right)=\omega(A^*A)+a\omega(A^*)+\overline{a}\omega(A)+|a|^2\|\omega\|\\\geq\omega(A^*A)-2|a|\omega(A^*A)^{\frac12}\|\omega\|^{\frac12}+|a|^2\|\omega\|=\left(\omega(A^*A)^{\frac12}-|a|\|\omega\|^{\frac12}\right)^2\geq0$$
Suppose it holds: $\lim\omega(E)=\|\omega\|$
Boundedness can be proven by an approximate identity: $$|\omega_E(A+a1)|=|\omega(A)+a\|\omega\||=\lim|\omega(AE)+a\omega(E)|\\\leq\limsup\|\omega\|\cdot\|aE+AE\|\leq\|\omega\|\cdot\|a1+A\|\cdot1$$ (Beware the limessuperior as the approximate identity alone diverges!)