Completely bounded mappings and simplicial complex structure in the primitive ideal space of a $C^*$-algebra
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- by Robert J. Archbold, Douglas W. B. Somerset and Richard M. Timoney PDF
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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.References
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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
- 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.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 1397-1427
- MSC (2000): Primary 46L05, 47B47, 46L06, 46L07
- DOI: https://doi.org/10.1090/S0002-9947-08-04666-7
- MathSciNet review: 2457404