The quotient embedding of tensor product

Question

Here is a quotation of a book:

Let $A, B$ be two $C^*$-algebras and $J\subset A$ be a $C^*$-subalgebra, then there is a dense embedding $$\frac{A\odot B}{J\odot B}\hookrightarrow\frac{A\otimes_{\max} B}{J\otimes_{\max} B}$$

Here, the $\odot$ denote the algebraic tensor product. However, what is this embedding? I mean for the element $x+J\odot B$ in $\frac{A\odot B}{J\odot B}$, what is the image of it in $\frac{A\otimes_{\max} B}{J\otimes_{\max} B}$? Or how to construct this embeding?

Answer 1

You map the product to the product. Barring any commentary otherwise, it's almost unthinkable that they could mean anything other than a homomorphism generated by the correspondence

$$ x \odot y \mapsto x \otimes_{\max} y $$