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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 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 ⊗ K(H)), with K(H) the compact operators, which requires us to develop the theory in the context of C*-algebras that are not necessarily unital.