Résume | We will describe Kirchberg's conjecture on tensor products of $C^*$-algebras which, as he showed, is equivalent to the Connes embedding problem. His conjecture is that the $C^*$-algebra of the free groups $C^*(\mathbb F_\infty)$ (or $C^*(\mathbb F_2)$) which has the Local Lifting Property (LLP in short) also has the Weak Expectation Property (WEP in short). We will describe the construction of the first example of a non-nuclear $ C^*$-algebra $A$ with both LLP and WEP. Our algebra $A$ has the ``same" finite dimensional operator spaces as $C^*(\mathbb F_\infty)$. In the second part several operator space variants involving analogues of the Gurarii space will be presented.
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