The Local Multiplier Algebra: Blending Noncommutative Ring Theory and Functional Analysis

  • Martin Mathieu
Conference paper
Part of the Trends in Mathematics book series (TM)


We discuss some basic features of the local multiplier algebra of a C*-algebra, the analytic analogue of the well-known Kharchenko-Martindale symmetric ring of quotients, and also the more recent maximal C*-algebra of quotients, which is the analytic companion to the Utumi-Lanning maximal symmetric ring of quotients, together with some of the applications to operator theory on C*-algebras. The emphasis lies in illustrating the interrelations between noncommutative ring theory and functional analysis.


Noncommutative rings of quotients C*-algebras AW*-algebras local multiplier algebra injective envelope operator modules derivations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    P. Ara, Some interfaces between noncommutative ring theory and operator algebras, Irish Math. Soc. Bull. 50 (2003), 7–26.zbMATHMathSciNetGoogle Scholar
  2. [2]
    P. Ara and M. Mathieu, A simple local multiplier algebra, Math. Proc. Cambridge Phil. Soc. 126 (1999), 555–564.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    P. Ara and M. Mathieu, Local multipliers of C*-algebras, Springer-Verlag, London, 2003.Google Scholar
  4. [4]
    P. Ara and M. Mathieu, A not so simple local multiplier algebra, J. Funct. Anal. 237 (2006), 721–737.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    P. Ara and M. Mathieu, Maximal C*-algebras of quotients and injective envelopes of C*-algebras, preprint.Google Scholar
  6. [6]
    K.I. Beidar, W.S. Martindale and A.V. Mikhalev, Rings with generalized identities, Marcel Dekker, New York, 1996.zbMATHGoogle Scholar
  7. [7]
    G.A. Elliott, Automorphisms determined by multipliers on ideals of a C*-algebra, J. Funct. Anal. 23 (1976), 1–10.zbMATHCrossRefGoogle Scholar
  8. [8]
    M. Frank and V.I. Paulsen, Injective envelopes of C*-algebras as operator modules, Pacific J. Math. 212 (2003), 57–69.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    K.R. Goodearl, Ring theory: Nonsingular rings and modules, Marcel Dekker, New York, 1976.zbMATHGoogle Scholar
  10. [10]
    V.K. Kharchenko, Automorphisms and derivations of associative rings, Kluwer Acad. Publ., Dordrecht, 1991.zbMATHGoogle Scholar
  11. [11]
    S. Lanning, The maximal symmetric ring of quotients, J. Algebra 179 (1996), 47–91.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    M. Mathieu, Elementary operators on prime C*-algebras, I, Math. Ann. 284 (1989), 223–244.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Mathieu, Derivations implemented by local multipliers, Proc. Amer. Math. Soc. 126 (1998), 1133–1138.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    M. Mathieu and A.R. Villena, The structure of Lie derivations on C*-algebras, J. Funct. Anal. 202 (2003), 504–525.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    E. Ortega, The maximal symmetric ring of quotients: Path algebras, incidence algebras and bicategories, PhD Thesis, Universitat Autònoma de Barcelona, Barcelona, 2006.Google Scholar
  16. [16]
    E. Ortega, Rings of quotients of incidence algebras and path algebras, J. Algebra 303 (2006), 225–243.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    V. Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Adv. Math. 78, Cambridge Univ. Press, Cambridge, 2002.Google Scholar
  18. [18]
    G.K. Pedersen, Approximating derivations on ideals of C*-algebras, Invent. math. 45 (1978), 299–305.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    D.W.B. Somerset, The proximinality of the centre of a C*-algebra, J. Approx. Theory 89 (1997), 114–117.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    D.W.B. Somerset, The local multiplier algebra of a C*-algebra, II, J. Funct. Anal. 171 (2000), 308–330.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2008

Authors and Affiliations

  • Martin Mathieu
    • 1
  1. 1.Department of Pure MathematicsQueen’s University BelfastBelfastNorthern Ireland

Personalised recommendations