  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 $C^*({\bb F}_2)$, 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 $C^*({\bb F}_2)$ or $C^*({\bb F}_\infty)$. 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
\emph{limit both inductive and projective} for a sequence of $C^*$-algebras
$(C_n)$ when each $C_n$ is a \emph{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.
