Q:

Why any operator is specified by this characteristic function?

On the paper "Tutorial Notes on One-Party and Two-Party Gaussian States", arXiv:quant-ph/0307196, the author states on section 2:

Any operator referring to a harmonic oscillator — position operator $q$, momentum operator $p$, both measured in natural units, so that $qp − pq = i$ — is a function of the familiar ladder operators $$a^\dagger=\dfrac{q-ip}{\sqrt{2}},\quad a=\dfrac{q+ip}{\sqrt{2}}.\tag{6}$$
We can specify such an operator $G(a^\dagger,a)$ by its characteristic function $C(z^\ast,z)$,
$$C(z^\ast,z)=\operatorname{Tr}\{e^{za^\dagger-z^\ast a}G(a^\dagger,a)\},\tag{7}$$
which is a numerical function of the complex phase space variables $$z^\ast=\dfrac{q'-ip'}{\sqrt{2}},\quad z=\dfrac{q'+ip'}{\sqrt{2}}.\tag{8}$$
Here, $q′, p′$ are the cartesian coordinates of classical phase space as one knows them from Hamilton’s approach to classical mechanics or the Liouville formulation of statistical mechanics.

Now, that any relevant operator for the harmonic oscilator can be written as a function $G(a^\dagger,a)$ of creation and annihilation operators, is something that I understand.
Now, why we can specify such an operator by the given function $C(z^\ast,z)$? I can't see why such function "encodes" the operator.
I believe this is some sort of Fourier transform, as later the author says that
$$G(a^\dagger,a)=\int\dfrac{dq'dp'}{2\pi}e^{-za^\dagger+z^\ast a}C(z^\ast,z).\tag{12}$$
But I can't see where all of this come from or why anyone would do it.
So where this $C(z^\ast,z)$ comes from, and what is the idea behind this construction?

A:

The characteristic, or generating, function in quantum mechanical systems is the noncommutative generalization of the corresponding concept of classical probability.
Let us consider the following classical situation (this can be generalized in many ways, but it is convenient to stick with a simple example here). Let $\mu$ be a probability (measure) acting on a finite dimensional real vector space $V$. Its characteristic function, or Fourier transform, $\hat{\mu}$ is defined as a function from the dual $V'$ of $V$ to the complex numbers as follows: for all $\omega\in V'$,
$$\hat{\mu}(\omega)=\int_V e^{i\omega(v)}\mathrm{d}\mu(v)\; .$$
The function $\hat{\mu}(\cdot)$ has the following properties: it is continuous, $\hat{\mu}(0)=\mu(V)=1$, and it is positive-defnite: for any $N\in\mathbb{N}$, $\{\alpha_i\}_{i=1}^N\subset \mathbb{C}$, and $\{\omega_i\}_{i=1}^N\subset V'$
$$\sum_{i,j=1}^N \alpha_i\bar{\alpha}_j \hat{\mu}(\omega_i-\omega_j)\geq 0\; .$$
Bochner's theorem actually tells us that 

There is a bijection between probabilities on $V$ and continuous functions on $V'$ that are positive-definite and have value one in zero; such bijection is exactly the Fourier transform.

Therefore the Fourier transform identifies uniquely (characterizes) a probability.
In quantum mechanics, there is a perfectly analogous noncommutative result. Let us consider the algebra of canonical commutation relations constructed over the finite dimensional real symplectic space $(S,\sigma)$. It is well-known that $(S,\sigma)\cong (\mathbb{R}^{2d},\omega)\cong (\mathbb{C}^d_{\mathbb{R}},\Im \langle \cdot,\cdot\rangle)$, where $\omega$ is the standard symplectic form, $\langle \cdot,\cdot\rangle$ the complex scalar product, and $\mathbb{C}^d_{\mathbb{R}}$ is the space $\mathbb{C}^d$ considered as a real vector space. In other words, it is possible to see the variables on which one constructs the algebra of canonical commutation relations as position and momentum $(q',p')\in \mathbb{R}^{2d}$ or as the complex variable $z\in \mathbb{C}^d$ (and its complex conjugate). 
The regular states of the algebra of canonical commutation relations are the states that can be written as density matrices in the usual Schrödinger representation. In other words, they are (positive) trace-class operators (of trace one) that depend only on the canonical quantum variables, i.e. the position and momentum operators or equivalently the creation and annihilation operators. These operators $\rho(a^*,a)$ are noncommutative probabilities in the quantum theory. Let me remark that since they are trace class, their trace can be taken and has a finite value, and they are positive operators. The fact that their trace is one is not important, and in fact everything could be done for positive trace class operators with arbitrary trace.
Let now $\rho(a^*,a)$ be a noncommutative probability. The role played by the character $e^{i\omega(v)}$ in a commutative theory is played by the Weyl operator $e^{a^*(z)-a(z)}$, $z\in \mathbb{C}^d$ in quantum mechanics. Therefore, it is natural to define the characteristic function, or noncommutative Fourier transform, in quantum mechanics as:
$$\hat{\rho}(z)=\mathrm{Tr}\{\,\rho(a^*,a)\, e^{a^*(z)-a(z)}\}\; .$$
$\hat{\rho}(z)$ is a complex number for any $z$ since $\rho(a^*,a)$ is trace class. In addition, it is a continuous function, $\hat{\rho}(0)=1$, and it is almost-positive-definite: for any $N\in\mathbb{N}$, $\{\alpha_i\}_{i=1}^N\subset \mathbb{C}$, and $\{z_i\}_{i=1}^N\subset \mathbb{C}^d$
$$\sum_{i,j=1}^N \alpha_i\bar{\alpha}_j \hat{\rho}(z_i-z_j)e^{i\Im \langle z_i,z_j\rangle}\geq 0\; .$$
It is very nice that for noncommutative probabilities, a noncommutative Bochner's theorem holds (proved by I. Segal in the fifties):

There is a bijection between regular states on the algebra of canonical commutation relations over $(S,\sigma)$ and continuous functions on $S$ that are almost-positive-definite and have value one in zero; such bijection is exactly the noncommutative Fourier transform.

Hence any regular quantum state (positive trace class operator) on the algebra of canonical commutation relations over $(S,\sigma)$ is characterized uniquely by a continuous and almost-positive-definite function on $S$. This is, in my opinion, a more precise version of the statement given by the authors of the paper cited by the OP. As a side remark, the noncommutative Bochner theorem is true also for bosonic quantum field theories, i.e. even if $S$ is infinite-dimensional (with suitable modifications).
As a final comment, if the function $\rho(a^*,a)$ is not positive, but still trace class, one should be a bit careful in giving its characteristic function. Every trace class operator $A$ can be uniquely written as the combination of four positive operators $A_1,A_2,A_3,A_4$:
$$A=A_1-A_2+i(A_3-A_4)\; .$$
Hence all the operators $\rho_1(a^*,a)$, $\rho_2(a^*,a)$, $\rho_3(a^*,a)$, $\rho_4(a^*,a)$ are characterized by their characteristic function with the usual properties, but the characteristic function of a non-positive $\rho(a^*,a)$ is not almost-positive-definite. Nonetheless, one may say that every trace class function of creation and annihilation operators is uniquely characterized by four continuous and almost-positive-definite functions on $S$.

