Completely bounded mappings and simplicial complex structure in the primitive ideal space of a $C^*$-algebra

We consider the natural contraction from the central Haagerup tensor product of a $C$*-algebra $A$ with itself to the space of completely bounded maps $CB(A)$ on $A$ and investigate those $A$ where there exists an inverse map with finite norm $L(A)$. We show that a stabilised version $L'(A) = \sup_n L(M_n(A))$ depends only on the primitive ideal space $\operatorname{Prim}(A)$. The dependence is via simplicial complex structures (defined from primal intersections) on finite sets of primitive ideals that contain a Glimm ideal of $A$. Moreover $L'(A) = L(A \otimes \mathcal{K}(H))$, with $\mathcal{K}(H)$ the compact operators, which requires us to develop the theory in the context of $C$*-algebras that are not necessarily unital.