At the beginning of Stinespring proof, Vern Paulsen considers the algebraic tensor product $\mathcal{A} \otimes \mathcal{H}$ of a (unital) $C^*$- Algebra $\mathcal{A}$ with the Hilbert space $\mathcal{H}$, with symmetric bilinear function
$$\langle a \otimes x, b \otimes y \rangle = \langle\phi(b^*a)x,y\rangle_{\mathcal{H}}$$
where $\phi : \mathcal{A} \rightarrow B(\mathcal{H})$ is a completely positive map. Now, I read about the algebraic tensor product for generic two vector spaces $V,W$, but I would like to understand better what '$\mathcal{A} \otimes \mathcal{H}$' actually means and also what $a \otimes x$ type objects are, in this context. Thank you
There are several ways to define tensor products. One that I like is to define, in your case, $a\otimes h$ as the linear map $a\otimes h: BL(A,H)\to \mathbb C$, where $BL(A,H)$ are the bilinear maps $A\times H\to\mathbb C$ and $$ (a\otimes h)\phi:=\phi(a,h). $$ Then $$A\otimes H=\operatorname{span}\{a\otimes h:\ a\in A,\ h\in H\}\subset L(BL(A,H),\mathbb C).$$
Another, less clear (in my opinion) but a bit less abstract way is to consider the formal span of elements of $A\times H$, and then quotient by all the relations that need to hold in the tensor product (like $\lambda (a\otimes h)=(\lambda a)\otimes h=a\otimes (\lambda h)$, bilinearity, etc.)