Toric Varieties written by David Cox, John Little and Hal Schenck p.18 Equivalence of constructions
Before starting our main result, we need to study the action of the torus $T_N$ on the semigroup algebra $C[M]$. The action of the torus $T_N$ on itself given by multiplication induces an action on $C[M]$ as follows: if $t \in T_N$ and $f \in C[M]$, then $t*f \in C[M]$ is defined by $p \mapsto f(t^{-1}*p)$ for $p \in V$. The minus sign will be explained in 5.0.
I cannot undestand this definition. What does this mean? My teacher said that the V in the passage above is a mistake.