Suppose that $A$ is a unital C$^\*$-algebra and that $\varphi: A \to \mathbb{C}$ is a bounded linear functional. Then there exists a Hilbert space $H$, a representation $\pi: A \to B(H)$ and vectors $\psi, \eta \in H$ such that $$\varphi(a) = \langle \pi(a)\psi, \eta \rangle$$
for all $a \in A$ (this can be proved by decomposing the functional as a linear combination of four states and considering the direct sum of the representation spaces associated to the GNS-construction for each state). 


My question is: 



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> Assuming further that $\| \varphi \| \leq 1$, can we choose $H$, $\pi$ and $\psi, \eta$ as above, **satisfying the additional requirement that $\| \psi \| \leq 1$ and $\| \eta \| \leq 1$**, such that (again) $\varphi(a) = \langle \pi(a) \psi, \eta \rangle$ for all $a \in A$?
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Note that this is clearly true for a positive functional - simply write $\varphi(a) = \langle \pi(a)\xi, \xi \rangle$ (using the GNS-construction) and note that $$ 1 \geq \| \varphi \| = \sup\_{\| a \| \leq 1} |\langle \pi(a) \xi, \xi \rangle | \geq |\langle \pi(1) \xi, \xi \rangle| = \| \xi \|^2$$



Bounded linear functionals and representations
Yes, you can do this using polar decomposition. We can also consider $\phi$ to be a normal linear functional on $A^{\*\*}$, and there is a positive $\omega \in A^\*$ and a partial isometry $v \in A^{\*\*}$ such that $\phi(a) = \omega(va)$ for all $x \in A$. (I'm sure this is in volume 1 of Takesaki, probably also in Pedersen.) We have $\|\omega\| = \|\phi\| \leq 1$, so we can apply GNS to $\omega$ and get $\phi(a) = \omega(va) = \langle \pi(va)\xi,\xi\rangle = \langle \pi(a)\psi,\eta\rangle$ with $\psi = \xi$ and $\eta = \pi(v^\*)\xi$. As you note, $\|\xi\|^2 = \langle \pi(1)\xi,\xi\rangle = \omega(\xi) \leq 1$.


