Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-08-04666-7
Published electronically: October 24, 2008
MathSciNet review: 2457404
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. S. D. Allen, A. M. Sinclair and R. R. Smith, The ideal structure of the Haagerup tensor product of $ C^*$-algebras, J. Reine Angew. Math. 442 (1993) 111-148. MR 1234838 (94k:46107)
  • 2. C. A. Akemann and G. K. Pedersen, Ideal perturbations of elements in $ C^*$-algebras, Math. Scand. 41 (1977) 117-139. MR 0473848 (57:13507)
  • 3. P. Ara and M. Mathieu, Local multipliers of $ C^*$-algebras, Springer-Verlag London, Ltd., London 2003. MR 1940428 (2004b:46071)
  • 4. R. J. Archbold, Topologies for primal ideals, J. London Math. Soc. (2) 36 (1987) 524-542. MR 918643 (89h:46076)
  • 5. R. J. Archbold and C. J. K. Batty, On factorial states of operator algebras III, J. Operator Theory 15 (1986) 53-81. MR 816234 (87h:46126)
  • 6. R. J. Archbold and E. Kaniuth, Simply connected nilpotent Lie groups with quasi-standard $ C^*$-algebras, Proc. Amer. Math. Soc. 125 (1997) 2733-2742. MR 1389503 (97j:22009)
  • 7. R. J. Archbold, E. Kaniuth, G. Schlichting and D. W. B. Somerset, Ideal spaces of the Haagerup tensor product of $ C^*$-algebras, Int. J. Math. 8 (1997) 1-29. MR 1433199 (97m:46089)
  • 8. R. J. Archbold, E. Kaniuth, G. Schlichting and D. W. B. Somerset, On the topology of the dual of a nilpotent Lie group, Math. Proc. Camb. Phil. Soc. 125 (1999) 269-293. MR 1643794 (2000b:22010)
  • 9. R. J. Archbold, J. Ludwig and G. Schlichting, Limit sets and strengths of convergence for sequences in the duals of thread-like Lie groups, Math. Zeit. 255 (2007) 245-282. MR 2262731 (2007j:22017)
  • 10. R. J. Archbold, D. W. B. Somerset and R. M. Timoney, On the central Haagerup tensor product and completely bounded mappings of a $ C^*$-algebra, J. Funct. Anal. 226 (2005) 406-428. MR 2160102 (2006d:46066)
  • 11. C. J. K. Batty and R. J. Archbold, On factorial states of operator algebras II, J. Operator Theory 13 (1985) 131-142. MR 768307 (86f:46065)
  • 12. J. Boidol, J. Ludwig and D. Müller, On infinitely small orbits, Studia Math. 88 (1988) 1-11. MR 932001 (89f:22009)
  • 13. A. Chatterjee and R. R. Smith, The central Haagerup tensor product and maps between von Neumann algebras, J. Funct. Anal. 112 (1993) 97-120. MR 1207938 (94c:46120)
  • 14. J. Dauns, The primitive ideal space of a $ C^*$-algebra, Canadian J. Math. 26 (1974) 42-49. MR 0336356 (49:1131)
  • 15. J. Dauns and K.-H. Hofmann, Representation of rings by sections, Memoirs of the American Mathematical Society, No. 83, American Mathematical Society, Providence 1968. MR 0247487 (40:752)
  • 16. J. Dixmier, Les $ C^*$-algèbres et leurs représentations, Gauthier-Villars, Paris 1964. MR 0171173 (30:1404)
  • 17. A. Guichardet, Tensor products of $ C^*$-algebras, (Part I, Finite tensor products), Aarhus University, Math. Institute, Lecture Notes Series No. 12, 1969.
  • 18. E. Kaniuth, G. Schlichting and K. F. Taylor, Minimal primal and Glimm ideal spaces of group $ C^*$-algebras, J. Funct. Anal. 130 (1995) 43-76. MR 1331977 (96g:46045)
  • 19. V. Paulsen, Completely bounded maps and operator algebras, Cambridge Univ. Press, Cambridge 2002. MR 1976867 (2004c:46118)
  • 20. I. Raeburn and D. P. Williams, Morita Equivalence and Continuous-Trace $ C*$-Algebras, Mathematical Surveys and Monographs Volume 60, American Mathematical Society, Providence 1998. MR 1634408 (2000c:46108)
  • 21. D. W. B. Somerset, Inner derivations and primal ideals of $ C$*-algebras, J. London Math. Soc. (2) 50 (1994) 568-580. MR 1299458 (95h:46107)
  • 22. D. W. B. Somerset, The central Haagerup tensor product of a $ C$*-algebra, J. Operator Theory 39 (1998) 113-121. MR 1610306 (99e:46073)
  • 23. R. M. Timoney, Some formulae for norms of elementary operators, J. Operator Theory 57 (2007) 121-145. MR 2301939 (2007m:47088)
  • 24. J. Tomiyama, Topological representations of $ C^*$-algebras, Tôhoku Math. J. (2) 14 (1962) 187-204. MR 0143053 (26:619)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L05, 47B47, 46L06, 46L07

Retrieve articles in all journals with MSC (2000): 46L05, 47B47, 46L06, 46L07


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: https://doi.org/10.1090/S0002-9947-08-04666-7
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.

American Mathematical Society