### 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 |
---|---|

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

**Completely bounded mappings and simplicial complex structure in the primitive ideal space of a C ^{*}-algebra.** / Archbold, Robert J; Somerset, Douglas W B; Timoney, Richard M.

Research output: Contribution to journal › Article

^{*}-algebra',

*Transactions of the American Mathematical Society*, vol. 361, no. 3, pp. 1397-1427. https://doi.org/10.1090/S0002-9947-08-04666-7

^{*}-algebra. Transactions of the American Mathematical Society. 2009 Mar;361(3):1397-1427. https://doi.org/10.1090/S0002-9947-08-04666-7

}

TY - JOUR

T1 - Completely bounded mappings and simplicial complex structure in the primitive ideal space of a C*-algebra

AU - Archbold, Robert J

AU - Somerset, Douglas W B

AU - Timoney, Richard M.

PY - 2009/3

Y1 - 2009/3

N2 - 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.

AB - 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.

KW - Haagerup tensor product

KW - operator-algebras

KW - factorial states

KW - primal ideals

KW - lie-groups

U2 - 10.1090/S0002-9947-08-04666-7

DO - 10.1090/S0002-9947-08-04666-7

M3 - Article

VL - 361

SP - 1397

EP - 1427

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 3

ER -