The norm on tensor product

Question

Here is a quotation of a book:

Let $B, ~C$ be unital C*-algebras and $A$ be a nonunital C*-algebra, $\|\cdot\|_{\alpha}$ be a C*-norm on $B\odot C$ (the tensor product) and $\|\cdot\|_{\beta}$ be the C*-norm on $A\odot C$ obtained by restricting $\|\cdot\|_{\alpha}$ to $A\odot C \subset B\odot C$.

Let $A_{1}=A+\mathbb{C}1_{B}$ and note that $A\odot C\subset B\otimes_{\alpha} C$. Hence $\|\cdot\|_{\beta}$ extends to a norm which yields an inclusion $A\otimes_{\beta} C\subset A_{1}\otimes_{\beta} C$. (Here, "$\otimes_{\alpha}$" denotes the completion of $\odot$ with the norm $\|\cdot\|_{\alpha}$).

I can not comprehend that " $\|\cdot\|_{\beta}$ extends to a norm which yields an inclusion $A\otimes_{\beta} C\subset A_{1}\otimes_{\beta} C$". Why can the $\|\cdot\|_{\beta}$ on $A\odot C$ be extended to $A_{1}\odot C$?

Answer 1

Because you can take $\|\cdot\|_\alpha$. This is a norm on $A_1\odot C$ that agrees with $\|\cdot\|_\beta$ on $A\odot C$.