From a proof that 2D Wightman CFT leads to a vertex algebra **[1]**:

Let $$ Y(a,z):=\frac{1}{(1+z)^{2\Delta_a}}\Phi_a\left(i\frac{1-z}{1+z}\right),\quad\text{with}\quad |z|<1. $$

Here $\Delta_a\ge 0$ is conformal weight of $\Phi_a$, and $\Phi_a$ is the scalar field satisfying the usual Wightman axioms (*Poincare covariance*, *stable vacuum* and *positive spectrum* of momentum, *completeness*, i.e. that polynomials of smeared fields applied on vacuum are dense in the Hilbert space, and *locality* (p. 6 of **[1]**)) and in addition, satisfying conformal covariance
$$
U(q,\Lambda,b)\Phi_a(x)U(q,\Lambda,b)^{-1}=(1+2x\cdot b+|x|^2 |b|^2)^{-\Delta_a}\Phi_a((q,\Lambda,b)\cdot x)
$$
with $(q,\Lambda,b)\mapsto U(q,\Lambda,b)$ being a unitary representation of the *conformal group*, fixing the vacuum vector $|0\rangle$.

On p. 11 of **[1]** the author writes:

We expand a chiral field $Y(a,z)$ in a Fourier series: $$ Y(a,z)=\sum_n a_{(n)}z^{-n-1},\quad\quad(\star) $$ where $a_{(n)}\in\mathrm{End}\; \mathcal{D}$.

Here $\mathcal{D}$ is the span of polynomials in the fields applied on the vacuum and is dense by *completeness* assumption.

I am not certain if this expansion is justified. Especially because it allows the author to prove that we indeed get a vertex algebra and hence we have an operator product expansion of the fields. Moreover, I found in another book **[2]** a similar proof on p. 190 and there the author writes

The operator product expansion (Axiom 6 on p. 168) of the primary fields allows to understand the fields $\Phi_a$ as fields $$ \Phi_a(z)=\sum a_{(n)} z^{-n-1} \in \mathrm{End} D [[z,z^{-1}]] $$ in the sense of vertex algebras.

Here $D$ is some dense domain in Hilbert space which is left invariant by fields, i.e. in **[2]** *completeness* is not assumed.

In other words, the author of **[2]** needs the existance of operator product expansion to write such a Fourier expansion.

My question:

$$ \text{is }(\star)\text{ true?} $$

References:

[1] V. Kac. *Vertex Algebras for Beginners*, 1998.

[2] M. Schottenloher. *A Mathematical Introduction to Conformal Field Theory*, 2008.