In most reference books about tensor product of $C^*$-algebras, the authors only give the definition of the spatial , maximal tensor product of unital $C^*$-algebras $A$ and $B$.
When $A$ and $B$ are non-unital, how to define the minimal tensor product and maximal tensor product? For example, in the non-unital case, can we define the maxiaml $C^*$ norm of $A\odot B$ as following?
$\|x\|_{max}=\sup_{\pi}\|\pi(x)\|$, where $\pi:A\odot B \to B(H)$ is a representation of $A\odot B$.