### Abstract

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

Original language | English |
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Pages (from-to) | 1397-1427 |

Number of pages | 31 |

Journal | Transactions of the American Mathematical Society |

Volume | 361 |

Issue number | 3 |

Early online date | 24 Oct 2008 |

DOIs | |

Publication status | Published - Mar 2009 |

### Keywords

- Haagerup tensor product
- operator-algebras
- factorial states
- primal ideals
- lie-groups

## Cite this

^{*}-algebra.

*Transactions of the American Mathematical Society*,

*361*(3), 1397-1427. https://doi.org/10.1090/S0002-9947-08-04666-7