Alternative proof of existence of absolute value of a functional on a C*-algebra The usual proof of the existence of an absolute value of a functional on a C*-algebra $A$ uses the polar decomposition of normal functionals on $A^{**}$, which relies on the compactness of the unit ball of $A^{**}$ in the weak*-topology.
Is it possible to derive the existence of an absolute value of a bounded linear functional on a C*-algebra via a compactness argument in $A^*$? For a definition of the absolute value of $\varphi \in A^*$, I mean a $\tau \in A^*_+$ such that $\|\tau\| = \|\varphi\|$ and $|\varphi(a)|^2 \leq \|\varphi\| \tau(a^* a)$.
Furthermore, is it possible to use this to show that $\varphi$ can be represented as $\varphi(a) = \langle \pi_\tau(a) \xi \,\vert\, \eta\rangle$ where $\pi_\tau$ is the GNS representation associated with $\tau$ and $\|\varphi\| = \|\xi\| \|\eta\|$?
 A: This is maybe a "backwards" answer to what you might have been hoping for...
An affirmative answer to the 2nd question would give the 1st question as well.  Indeed, 
if there is a $*$-representation $\pi:A\rightarrow B(H)$ and $\xi,\eta\in H$ with $\|\varphi\| = \|\xi\| \|\eta\|$ and $\varphi(a) = \langle \pi(a)\xi, \eta \rangle$, then by rescaling we may suppose that $\|\xi\|=\|\eta\|=\|\varphi\|^{1/2}$, and then $\tau:a\mapsto \langle \pi(a)\xi,\xi\rangle$ is a positive functional with $\|\tau\| = \|\xi\|^2 = \|\varphi\|$.   Further,
$$ \|\varphi\| \tau(a^*a) = \|\eta\|^2 \|\pi(a)\xi\|^2
\geq | \langle \pi(a)\xi, \eta \rangle|^2 = |\varphi(a)|^2. $$

I do not know a "compactness" argument which can show this.  However, a while ago I wrote up an argument of how to prove Kaplansky Density using Arens products; see Notes on GitHub.  To do this in a non-circular way, you need to use Hahn-Banach, and need to find a way to prove exactly this representation result, in a "simple" way.  The way I did this was as follows.
Let $H$ be a Hilbert space.  Let $B(H)_*$ be the trace-class operators on $H$, the predual of $B(H)$.  By Hahn-Banach (Goldstine's Theorem) for $\mu\in B(H)^*$ there is a net $(\omega_i)$ in $B(H)_*$ converging weak$^*$ to $\mu$, and with $\|\omega_i\|=\|\mu\|$ for each $i$.  Thus, for an ultrafilter $\mathcal U$ refining the order filter, the natural map $(B(H)_*)_{\mathcal U} \rightarrow B(H)^*$ (given by "take weak$^*$-limit") is a metric surjection.  With $K = \ell^2(H)$, and $\pi_0:B(H)\rightarrow B(K)$ the "diagonal" map, for each $\omega\in B(H)_*$ we can find $\xi,\eta\in K$ with $\omega = \omega_{\xi,\eta}\circ\pi_0$.  Now let $K = (\ell^2(H))_{\mathcal U}$ the ultrapower, a Hilbert space.  We can find $\xi = (\xi_i), \eta=(\eta_i)\in K$ with $\omega_i = \omega_{\xi_i,\eta_i}\circ\pi_0$ for each $i$.  Thus, with $\Pi:B(H)\rightarrow B(K)$ the diagonal of $\pi_0$, we have that
$$ \langle \Pi(x)(\xi), \eta \rangle = \lim_{i\rightarrow\mathcal U} \langle \pi_0(x)\xi_i, \eta_i \rangle = \lim_{i\rightarrow\mathcal U} \omega_i(x) = \mu(x). $$
Given a $C^*$-algebra $A$, by the GNS construction, we can exhibit $A$ as a subalgebra of $B(H)$ for some $H$.  For $\varphi\in A^*$ take a Hahn-Banach extension to $\mu\in B(H)^*$.  Then the previous paragraph, with $\pi$ the restriction of $\Pi$ to $A$, gives the required representation.
