# Strengthening of Connes' embedding conjecture

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?

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.