I am reading about affine toric varieties from Toric Varieties by Cox, Little, and Shenck, and the definition of an affine toric variety is given as follows:
An affine toric variety is an irreducible affine variety $V$ containing a torus $T_N \simeq (\mathbb{C}^\times)^n$ as a Zariski open subset such that the action of $T_N$ on itself extends to an algebraic action of $T_N$ on $V$.
This is first time I am reading this and I don't understand which action are they talking about. Are they saying any action between $T_N$ to $T_N$ will be extended to an algebraic action? It will be great if you can help me with this doubt.