with $z_{0}=0$.) This proves that $Y_{W}^{(z_{0})}$ satisfies the opposite Jacobi identity. $\;\;\;\;\Box$

Notice that for a $V$-module $W$, $W'$ is considerably smaller than $W^{*}$ if $W$ is infinite-dimensional — This turns out to be the case almost all of the times. Now we consider the whole space $W^{*}$. Clearly, the definition (\[ey’\]) still makes sense if one replaces $w'$ (an element of $W'$) by any element of $W^{*}$. Let $W$ be a weak $V$-module for now and let $v\in V,\;\alpha\in W^{*}$. We define $$\begin{aligned}
Y^{*}(v,x)\alpha\in W^{*}[[x,x^{-1}]]\end{aligned}$$ by $$\begin{aligned}
\label{e*1}
\<Y^{*}(v,x)\alpha, w\>=\<\alpha,Y^{o}(v,x)w\>\end{aligned}$$ for $w\in W$.

By Proposition 5.3.1 of \[FHL\] we have $$\begin{aligned}
\label{e*2}
\<\alpha,Y(v,x)w\>
=\left\<Y^{*}\left(e^{xL(1)}\left(-x^{-2}\right)^{L(0)}v,x^{-1}\right)\alpha,w
\right\>.\end{aligned}$$ Since $Y^{o}({\bf 1},x)=1$ (recall (\[yo1\])), we have $$\begin{aligned}
\label{ey*1}
Y^{*}({\bf 1},x)=1.\end{aligned}$$ For $v\in V$, we set $$\begin{aligned}
Y^{*}(v,x)=\sum_{n\in {\Z}}v^{*}_{n}x^{-n-1}.\end{aligned}$$ When $v=\omega$ (the Virasoro element), we set $$\begin{aligned}
Y^{*}(\omega,x)=\sum_{n\in {\Z}}L^{*}(n)x^{-n-2},\end{aligned}$$ i.e., $L^{*}(n)=\omega^{*}_{n+1}$ for $n\in {\Z}$. Then (by (\[eyomega\])) $$\begin{aligned}
\label{eL*n}
\< L^{*}(n)\alpha,w\>=\<\alpha, L(-n)w\>\end{aligned}$$ for $n\in {\Z},\; \alpha\in W^{*},\; w\in W$.

The opposite Jacobi identity (\[ehl\]) directly gives the following Jacobi identity in terms of matrix-coefficients: $$\begin{aligned}
\label{ejacobiy*m}
& &x_{0}^{-1}\delta\left(\frac{x_{1}-x_{2}}{x_{0}}\right)
\<Y^{*}(u,x_{1})Y^{*}(v,x_{2})\alpha,w\>\nonumber\\
& &-x_{0}^{-1}\delta\left(\frac{x_{2}-x_{1}}{-x_{0}}\right)
\<Y^{*}(v,x_{2})Y^{*}(u,x_{1})\alpha,w\>\nonumber\\
&=&x_{2}^{-1}\delta\left(\frac{x_{1}-x_{0}}{x_{2}}\right)
\<Y^{*}(Y(u,x_{0})v,x_{2})\alpha,w\>\end{aligned}$$ for $u,v\in V,\; \alpha\in W^{*},\;w\in W$.

As an immediate consequence we have (cf. \[Li5\]):

For any weak $V$-module $W$, $W^{*}$ 