Convex combinations of product states

Question

Is it true that the set of convex combinations of product states on a tensor product of C-algebras is weak-dense in the set of all states on said tensor product?

Answer 1

No. Not even in the finite-dimensional case.

Let $A=B=M_2(\mathbb C)$. We can identify $$\tag1 A\otimes B\simeq M_4(\mathbb C)$$ via the Kronecker product.

Note that we don't need to be careful about the topology here, since these are finite-dimensional algebras and so they have a unique locally convex topology.

The states on $M_2(\mathbb C)$ are of the form $\operatorname{Tr}(H\cdot)$, with $H$ positive and trace one. Then $$\tag2 \operatorname{Tr}(H\,\cdot\,)\otimes\operatorname{Tr}(K\,\cdot\,)=\operatorname{Tr}((H\otimes K)\,\cdot\,) $$ (the uniqueness of the trace up to a scalar multiple guarantees the equality $(2)$. So the question is whether the positive elements of the form $\sum_j t_j(a_j\otimes b_j)$ with $a_j,b_j\geq0$ and $t_j$ convex coefficients, are dense in the positive elements in $M_4(\mathbb C)$. As the state space in a finite-dimensional C$^*$-algebra is compact, and the convex hull of compact sets is compact in finite dimension, density here means equality. And the answer is no, even if we consider sums and not just convex combinations.