If $A$ and $B$ are $C^\*$ algebras I will write $A \overline{\otimes} B$ for the maximal tensor product and $A \underline{\otimes} B$ for the minimal tensor product. If $G$ is a countable discrete group I will write $C^\*(G)$ for the full group $C^\*$ algebra of $G$. Let also $\mathbb{F}\_\infty$ be the free group on a countably infinite set of generators.


A theorem of Kirchberg asserts that Connes' embedding conjecture is equivalent to the statement $C^\*(\mathbb{F}\_\infty) \overline{\otimes} C^\*(\mathbb{F}\_\infty) = C^\*(\mathbb{F}\_\infty) \underline{\otimes} C^\*(\mathbb{F}\_\infty)$. I would like to find out what is known about the statement that $C^\*(\mathbb{F}\_\infty) \overline{\otimes} C^\*(G) = C^\*(\mathbb{F}\_\infty) \underline{\otimes} C^\*(G)$ for every countable discrete group $G$. Are there counterexamples to this stronger statement, or alternatively is it equivalent to the original conjecture?



Strengthening of Connes' embedding conjecture
Your more general conjecture is not true. This property for a discrete group $G$ is equivalent to the fact that $G$ is WEP (its $C^{\ast}$-algebra has weak expectation property). In "Examples of hyperlinear groups without factorization property" Andreas Thom constructs a (hyperlinear) property (T) group that is not residually finite. On the other hand, by a result of Kirchberg ("Discrete groups with Kazhdan's property T and factorization property are residually finite") such groups cannot have WEP.


