is a $g(V)$-module where $v(n)$ acts as $v_{n}^{*}$ for $v\in V, n\in {\Z}$. In particular, the following commutator formula (cf. (\[ecomm\])) holds: $$\begin{aligned}
[u_{m}^{*},v_{n}^{*}]=\sum_{i\ge 0}{m\choose i}(u_{i}v)_{m+n-i}^{*}\end{aligned}$$ for $u,v\in V,\; m,n\in {\Z}$. $\;\;\;\;\Box$

[*From now on we shall freely use the commutator formula on $W^{*}$ without explicit comments.*]{}

The following definition naturally arises from Remark \[rproof\] (cf. \[Li5\]):

As it has been noticed in \[Li5\], from (\[ehl\]) we immediately have:

The pair $(D(W),Y^{*})$ carries the structure of a weak $V$-module and $D(W)$ is the unique maximal weak $V$-module in $W^{*}$ with $Y^{*}$ being the vertex operator map.$\;\;\;\;\Box$

If $W=\coprod_{h\in {\C}}W_{(h)}$ is an ordinary $V$-module, then $W^{*}=\prod_{h\in {\C}}W_{(h)}^{*}$ is a formal completion of $W'$ and $Y^{*}$ is the natural extension of $Y'$ (cf. \[HL1\]). Let $W$ be a $V$-module and let $\alpha\in W^{*}$. Then from (\[eL\*n\]), $\alpha$ is an eigenvector of $L^{*}(0)$ with eigenvalue $h$ if and only if $\alpha\in W_{(h)}^{*}\subset W'$. Therefore, $W'$ is the largest generalized submodule of $D(W)$. Let $\cal{A}$ be the class of vertex operator algebras satisfying the condition that $L(0)$ acts semisimply on every weak module, i.e., every weak module is a generalized module. Then we immediately have (cf. \[Li5\]):

If $V$ is of class ${\cal{A}}$ and 