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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 )o nA and investigate those A where there exists an inverse map with finite norm L(A). We show that a stabilised version L (A )= supn L(Mn(A)) depends only on the primitive ideal space Prim(A). The dependence is via simplicial complex structures (defined from primal inter- sections) 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.

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