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 $\|\varphi\| = \|\xi\| \|\eta\|$?



Alternative proof of existence of absolute value of a functional on a C*-algebra
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,
$$ \|\varphi\| \tau(a^\*a) = \|\eta\|^2 \|\pi(a)\xi\|^2
\geq | \langle \pi(a)\xi, \eta \rangle|^2 = |\varphi(a)|^2. $$




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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](https://github.com/MatthewDaws/Mathematics/tree/master/Kaplansky). 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.


