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

Robert J. Archbold, Douglas W. B. Somerset, Richard M. Timoney
2008 Transactions of the American Mathematical Society  
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
more » ... tain 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.
doi:10.1090/s0002-9947-08-04666-7 fatcat:5jnd4oaszzhd7nope6pz6kgeci