Definition of 'product states' for $C^*$-algebras I am slightly confused by the definition of 'product states' for $C^*$-algebras. So far I have the following:
Let $A$ and $B$ be $C^*$-algebras, $\varphi\in\mathcal{S}(A)$, $\psi\in\mathcal{S}(B)$. Then $\varphi\odot\psi$ is a positive linear functional on $A\odot B$. But at this point there is no $C^*$-norm defined on the $*$-algebra $A\odot B$, so I am at a loss how to establish that the 'product state' is indeed a state on $A\odot B$ i.e. that its norm is $1$.
Also I have seen in a proof reference to a function from 'product states of $A\odot B$ to product states of $C\odot D$'. Is every state on $A\odot B$ (for a given $C^*$-norm) a product of states on $A$ and $B$?
Also is there is any good reference where I can read more about this?
 A: 1. Tensor product of $C^*$-algebras
A  $C^*$-algebra $A$ is called nuclear if, for every other $C^*$-algebra $B$, there is a unique $C^*$-norm on
$A\odot B$, whose completion is denoted   $A\otimes B$, and called the $C^*$-tensor product of $A$ and $B$.
Every commutative $C^*$-algebra is nuclear, and so are AF-algebras, and a large number of $C^*$-algebras one encounters
in applications.  A notable exception  is the group $C^*$-algebra of the free group on two generators.
If neither $A$ or $B$ are nuclear, there are two important $C^*$-norms on   $A\odot B$, respectively denoted
$\|\cdot\|_{\text{min}}$ and $\|\cdot\|_{\text{max}}$, and any other $C^*$-norm sits in between these two.
2. Product states
If $A_1$ and $A_2$ are $C^*$-algebras equipped with states $\varphi _1$ and $\varphi _2$, then one may look at the GNS representation
$(\pi _i,H_i,\xi _i)$ for each $\varphi _i$.  The algebraic tensor product   $A_1\odot A_2$ may then be represented on the Hilbert
space $H_1\otimes H_2$ via a $^*$-representation  $\pi $ such that
$$
  \pi (a\odot b) = \pi _1(a)\otimes \pi _2(b).
  $$
The vector state $\varphi $ associated to the vector $\xi _1\otimes \xi _2$, namely
$$
  \varphi (c) =   \langle \pi (c)(\xi _1\otimes \xi _2),\xi _1\otimes \xi _2\rangle , \quad \forall c\in   A_1\odot A_2,
  $$
then satisfies
$$
  \varphi (a\odot b) =
  \langle \pi (a\odot b)(\xi _1\otimes \xi _2),\xi _1\otimes \xi _2\rangle  = $$$$ =
  \langle \pi _1(a)\xi _1,\xi _1\rangle    \langle \pi _2(b)\xi _2,\xi _2\rangle  =
  \varphi _1(a)\varphi (b),
  $$
so we see that $\varphi $ is the product state of $\varphi _1$ and $\varphi _2$.
It can be shown that $\pi $ is continuous wrt $\|\cdot\|_{\text{min}}$ and hence also with respect to any $C^*$-norm on
$A_1\odot A_2$,  because,  as already mentioned,  $\|\cdot\|_{\text{min}}$ is the smallest of all $C^*$-norms.
As a consequence, the product state is also continuous with respect to any $C^*$-norm on
$A_1\odot A_2$.
Every vector state associated to a vector of norm 1 (such as $\xi _1\otimes \xi _2$) is a state (i.e. has morm 1), so the product state is indeed a state.

Regarding references, I'd say Murphy's book on C*-algebras is a good starting point.
