The extended Hall torus action is well-defined

Let $x,y$ be homogeneous of degrees $\beta ,\gamma$. By [F1], $\mathrm{deg}(x\star y)=\beta +\gamma$. Thus ${\nu }^{(\alpha ,\beta +\gamma )}(x\star y)={\nu }^{(\alpha ,\beta )+(\alpha ,\gamma )}(x\star y)$ by [F2], which is the product of the two scaled factors. Therefore the scaling preserves multiplication.

Scaling by ${\nu }^{-(\alpha ,\mathrm{deg}x)}$ is the inverse map, so the scaling in step 1.1 is an automorphism. Associativity of the crossed product follows from [F3], the action law in step 1.1, and addition in $K_0$: in the declared left-torus normal order, for homogeneous $x,y,z$ of degrees $m,n,p$, the product is $(k_{\alpha }x)(k_{\beta }y)={\nu }^{-(\beta ,m)}{k}_{\alpha +\beta }(x\star y)$. A triple $k_{\alpha }x,k_{\beta }y,k_{\gamma }z$ contributes exponent $-(\beta ,m)-(\gamma ,m+n)$ in the first parenthesization and $-(\gamma ,n)-(\beta +\gamma ,m)$ in the second. Bilinearity makes these equal, and [F3] identifies the Hall products. The unit is $k_0[0]$, and the conjugation relation follows by the same product formula.