This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.

Consider the following objects:

• the group $$\mathrm{GL}_{\infty}=\{A=(a_{ij})_{i,j\in\mathbb{Z}}|A\text{ is invertible and all but a finite number of }a_{ij}-\delta_{ij}\text{ is }0\}$$
• the infinite wedge space $F^{(0)}V$ with $V=\mathbb{C}^{\infty}$ and $$F^{(0)}V=\{v_{i_0}\wedge v_{i_{-1}}\wedge v_{i_{-2}}\cdots|i_0>i_{-1}>\cdots\text{ and }i_{-k}=-k\text{ for }k>>0\}$$

There is an action of $\mathrm{GL}_{\infty}$ on $\mathbb{C}^{\infty}$ given by $$A.(v_{i_0}\wedge v_{i_{-1}}\wedge v_{i_{-2}}\wedge\cdots) =Av_{i_0}\wedge Av_{i_{-1}}\wedge Av_{i_{-2}}\wedge\cdots.$$

Then a larger group $\overline{\mathrm{GL}}_{\infty}$ is considered: $$\overline{\mathrm{GL}}_{\infty}=\{A=(a_{ij})_{i,j\in\mathbb{Z}}|A\text{ is invertible and all but a finite number of }a_{ij}-\delta_{ij}\text{ with }i\geqslant j\text{ is }0\}$$

It is claimed in the section 6.2 of the book that the action of $\mathrm{GL}_{\infty}$ on $F^{(0)}V$ is extended to an action of $\overline{\mathrm{GL}}_{\infty}$ on $V$. But I find the natural way to extend this action does not work.

As an example, take $A\in\overline{\mathrm{GL}}_{\infty}$ given by $$a_{ii}=1,\quad a_{i-1,i}=2,\text{ and } a_{ij}=0\text{ for all other }(i,j)$$ Then $$Av_{0}\wedge Av_{-1}\wedge Av_{-2}\wedge\cdots=(v_{0}+2v_{-1})\wedge (v_{-1}+2v_{-2})\wedge (v_{-2}+2v_{-3})\wedge\cdots$$ which does not make sense. So how should I modify the action to obtain a well-defined action of $\overline{\mathrm{GL}}_{\infty}$ on $F^{(0)}V$?