A non-nuclear $C^*$-algebra with the weak expectation property and the local lifting property

Abstract

We construct the first example of a $C^*$-algebra A with the properties in the title. This gives a new example of non-nuclear A for which there is a unique $C^*$-norm on $A \otimes A^{op}$. This example is of particular interest in connection with the Connes–Kirchberg problem, which is equivalent to the question whether $C^*({\mathbb {F}}_2)$, which is known to have the LLP, also has the WEP. Our $C^*$-algebra A has the same collection of finite dimensional operator subspaces as $C^*({\mathbb {F}}_2)$ or $C^*({\mathbb {F}}_\infty )$. In addition our example can be made to be quasidiagonal and of similarity degree (or length) 3. In the second part of the paper we reformulate our construction in the more general framework of a $C^*$-algebra that can be described as the limit both inductive and projective for a sequence of $C^*$-algebras $(C_n)$ when each $C_n$ is a subquotient of $C_{n+1}$. We use this to show that for certain local properties of injective (non-surjective) $*$-homomorphisms, there are $C^*$-algebras for which the identity map has the same properties as the $*$-homomorphisms.