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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Authors: Robert J. Archbold, Douglas W. B. Somerset and Richard M. Timoney
Journal: Trans. Amer. Math. Soc. 361 (2009), 1397-1427
MSC (2000): Primary 46L05, 47B47, 46L06, 46L07
Published electronically: October 24, 2008
MathSciNet review: 2457404
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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 $ \operatorname{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 \otimes \mathcal{K}(H))$, with $ \mathcal{K}(H)$ the compact operators, which requires us to develop the theory in the context of $ C$*-algebras that are not necessarily unital.


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Additional Information

Robert J. Archbold
Affiliation: Department of Mathematical Sciences, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland, United Kingdom
Email: r.archbold@maths.abdn.ac.uk

Douglas W. B. Somerset
Affiliation: Department of Mathematical Sciences, University of Aberdeen, King’s College, Aberdeen AB24 3UE, Scotland, United Kingdom
Email: somerset@quidinish.fsnet.co.uk

Richard M. Timoney
Affiliation: School of Mathematics, Trinity College, Dublin 2, Ireland
Email: richardt@maths.tcd.ie

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04666-7
PII: S 0002-9947(08)04666-7
Received by editor(s): February 1, 2007
Published electronically: October 24, 2008
Additional Notes: The work of the third author was supported in part by the Science Foundation Ireland under grant 05/RFP/MAT0033.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.