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From Blumenhagen and Plauschinn Introduction to Conformal Field Theory, 2009:

"The proof that the OPE of two quasi-primary fields involves indeed just other quasi-primary fields and their derivatives is non-trivial and will not be presented".

Does anybody know a good reference with a proof of this statement?

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    $\begingroup$ Any state in a quantum theory can be expanden into a complete basis of energy eigenstates. In a radially quantized CFT the hamiltonian is the dilation operator so the eigenstates are states of definite scaling dimension. Given the commutators of the conformal algebra one can increase the dimension by applying the momentum op. (derivatives). So it's not too strange that all eigenstates will be quasi-primaries and their descendants. $\endgroup$
    – Prastt
    Commented Nov 22, 2015 at 1:50

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Here we consider the case of unitary highest weight representations: there is a primary state $\left|\phi\right>$ and the representation $V$ is spanned by $L_{-k_1}\dots L_{-k_n}\left|\phi\right>$. Then the quasi-primary states are those annilated by $L_1$ and the derivative is $L_{-1}$.

Let $W=\ker(L_1)$ our claim is $V=W\oplus L_{-1}W\oplus L_{-1}^2W\oplus \dots$. To prove this we make use of grading: $L_{-k_1}\dots L_{-k_n}\left|\phi\right>$ has grade $k_1+\dots+k_n$. So $L_1$ decrease grading by $1$ and $L_{-1}$ increase grading by $1$.

So look at the $n$ part $V_n$, and treat $L_1$ be an operator from $V_n$ to $V_{n-1}$, and $L_{-1}$ be an operator from $V_{n-1}$ to $V_{n}$. with induction we only need to prove $V_n=\ker(L_1)\oplus\operatorname{Img}(L_{-1})$. Which follows from $L_{-1}=L_1^{\dagger}$.

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