if $(z-w)^N
[a(z),b(w)]=0$ for some $N\geq 0$. Here $[,]$ denotes the super bracket. This condition implies that $a\circ_n b = 0$ for $n\geq N$, so (\[opeform\]) becomes a finite sum. A [*commutative quantum operator algebra*]{} (CQOA) is a QOA whose elements pairwise quantum commute. Finally, the notion of a CQOA is equivalent to the notion of a vertex algebra. Every CQOA $\cA$ is itself a faithful $\cA$-module, called the [*left regular module*]{}. Define $$\rho:\cA\rightarrow QO(\cA),\ \ \ \ a\mapsto\hat a,\ \ \ \ \hat
a(\zeta)b=\sum_{n\in\mathbb{Z}} (a\circ_n b)~\zeta^{-n-1}.$$ Then $\rho$ is an injective QOA homomorphism, and the quadruple of structures $(\cA,\rho,1,\partial)$ is a vertex algebra in the sense of [@FLM]. Conversely, if $(V,Y,{\bf 1},D)$ is a vertex algebra, the collection $Y(V)\subset QO(V)$ is a CQOA. [*We will refer to a CQOA simply as a vertex algebra throughout the rest of this paper*]{}.

The following are useful identities that measure the nonassociativity and noncommutativity of the Wick product, and the failure of the positive circle products to be derivations of the Wick product. Let $a,b,c$ be vertex operators in some vertex algebra $\cA$, and let $n > 0$. Then $$\label{vaidi} :(:ab:)c:-:abc:=\sum_{k\geq0}{1\over(k+1)!}\left(:(\partial^{k+1}a)(b\circ_k
c): +(-1)^{|a||b|}:(\partial^{k+1}b)(a\circ_k c):\right),$$ $$\label{vaidii} :ab:-(-1)^{|a||b|}:ba:=\sum_{k\geq0}{(-1)^k\over(k+1)!}\partial^{k+1}(a\circ_kb),$$ $$\label{vaidiii} a\circ_n(:bc:)-:(a\circ_nb)c:-(-1)^{|a||b|}:b(a\circ_nc):= \sum_{k=1}^n\left(\begin{matrix} n\cr k\end{matrix} \right)(a\circ_{n-k}b)\circ_{k-1}c,$$ 