Skip to main content
SpringerLink
Log in

Single Elements of Operator Algebras

  • Conference paper
Book cover Singular Integral Operators, Factorization and Applications

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 142))

  • 295 Accesses

Abstract

An element sof an (abstract) algebra Ais asingle elementofAif asb =0 and ab∈A implies thatas =0 orsb =0. Such elements were first used by operator theorists in 1961 when J.R. Ringrose proved that algebraic isomorphisms between certain nest algebras were spatial. They have been successfully used since, towards a similar end, namely, in proving the (quasi-)spatiality of algebraic isomorphisms between reflexive operator algebras. These results, and others concerning the structure and the rank of single elements in operator algebras of various types, are reviewed in this article.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Dalla, L., Giotopoulos, S., Katseli, N., Skeletons of the unit ball of a C*-algebraMath. Balkanica (N. S.)(1)1(1987), 83–88.

    MathSciNet  MATH  Google Scholar 

  2. Dalla, L., Giotopoulos, S., Katseli, N., The socle and finite-dimensionality of a semiprime Banach algebra, Studia Math. (2) 92 (1989), 201–204.

    MathSciNet  MATH  Google Scholar 

  3. Erdos, J.A., On certain elements ofC*algebrasIllinois J. Math.15 (1971), 682–693.

    MathSciNet  MATH  Google Scholar 

  4. Erdos, J.A., Giotopoulos, S., Lambrou, M.S., Rank one elements of Banach algebrasMathematika24 (1977), 178–181.

    Article  MathSciNet  Google Scholar 

  5. Gilfeather, F., Moore, R.L., Isomorphisms of certain CSL algebrasJ. Funct. Anal. (2) 67 (1986), 264–291.

    Article  MathSciNet  MATH  Google Scholar 

  6. Giotopoulos, S., A note on annihilator Banach algebrasMath. Proc. Cambridge Philos. Soc.97 (1985), 101–106.

    Article  MathSciNet  MATH  Google Scholar 

  7. Giotopoulos, S., Cr-classes of operators in C*-algebrasJ. Austral. Math. Soc. Ser. A(2) 40 (1986), 153–160.

    Article  MathSciNet  MATH  Google Scholar 

  8. Giotopoulos, S., Katseli, N., Modular annihilator algebrasPeriod. Math. Hangar. (2) 22 (1991), 91–95.

    Article  MathSciNet  MATH  Google Scholar 

  9. Harrison, K.J., Longstaff, W.E., Automorphic images of commutative subspace latticesTrans. Amer. Math. Soc. (1) 296 (1986), 217–228.

    Article  MathSciNet  MATH  Google Scholar 

  10. Katavolos, A., Lambrou, M.S., Longstaff, W.E., Pentagon subspace lattices on Ba-nach spacesJ. Op. Theory46 (2001), 355–380.

    MathSciNet  MATH  Google Scholar 

  11. Lambrou, M.S., Automatic continuity and implementation of homomorphisms, U.Crete (manuscript)(1982).

    Google Scholar 

  12. Lambrou, M.S., On the ranks of operators in reflexive algebrasLin. Alg. & Applic.142 (1990), 211–235.

    Article  MathSciNet  MATH  Google Scholar 

  13. Longstaff, W.E., Strongly reflexive latticesJ. London Math. Soc. (2) 11 (1975), 491–498.

    Article  MathSciNet  MATH  Google Scholar 

  14. Longstaff, W.E., Panaia, O., On the ranks of single elements of reflexive operator algebrasProc. Amer. Math. Soc. (10) 125 (1997), 2875–2882.

    Article  MathSciNet  MATH  Google Scholar 

  15. Longstaff, W.E., Panaia, O., J-subspace lattices and subspace M-bases, Studia Math. (3) 139 (2000), 197–212.

    MathSciNet  MATH  Google Scholar 

  16. Longstaff, W.E., Panaia, O., Single elements of matrix incidence algebrasLin. Alg. & Applic.318 (2000), 117–126.

    Article  MathSciNet  MATH  Google Scholar 

  17. Longstaff, W.E., Panaia, O., Single elements of finite CSL algebrasProc. Amer. Math. Soc.(4) 129 (2000), 1021–1029.

    Article  MathSciNet  Google Scholar 

  18. Longstaff, W.E., Nation, J.B., Panaia, O., Abstract refexive sublattices and completely distributive collapsibilityBull. Aust. Math. Soc.58 (1998), 245–260.

    Article  MathSciNet  MATH  Google Scholar 

  19. Moore, R.L., Isomorphisms of certain CSL algebrasU. Alabama (manuscript)(1983),.

    Google Scholar 

  20. Munn, W.D., Brandt congruences on inverse semigroups, Proc. London Math. Soc. (3) 14 (1964), 154–164.

    Article  MathSciNet  MATH  Google Scholar 

  21. Panaia, O., Quasi-spatiality of isomorphisms for certain classes of operator algebrasPh. D. dissert. U. West. Aust.(1995).

    Google Scholar 

  22. Panaia, O., Algebraic isomorphisms and finite distributive subspace lattices, J. London Math. Soc. (2) 59 (1999), 1033–1048.

    Article  MathSciNet  MATH  Google Scholar 

  23. Radjavi, H., Rosenthal, P., Invariant subspaces and weakly closed algebrasBull. Amer. Math. Soc.(5) 74 (1968), 1013–1014.

    Article  MathSciNet  MATH  Google Scholar 

  24. Ringrose, J.R., Algebraic isomorphisms between ordered bases, Amer.J. Math.83 (1961), 463–478.

    Article  MathSciNet  MATH  Google Scholar 

  25. Ringrose, J.R., On some algebras of operators IIProc. London Math. Soc.(3) 16 (1966), 385–402.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics & Statistics, The University of Western Australia 35 Stirling Highway, Crawley Western Australia, 6009, Australia

    William E. Longstaff

Authors
  1. William E. Longstaff
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, Technical University Chemnitz, Chemnitz, 09107, Germany

    Albrecht Böttcher

  2. Department of Mathematics and Computer Science, Vrije Universiteit Amsterdam, De Boelelaan, Amsterdam, NL-1081 HV, The Netherlands

    Marinus A. Kaashoek

  3. Departamento de Matemática, Instituto Superior Técnico, Ave. Rovisco Pais, Lisboa, 1049-001, Portugal

    Amarino Brites Lebre , António Ferreira dos Santos  & Frank-Olme Speck ,  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer Basel AG

About this paper

Cite this paper

Longstaff, W.E. (2003). Single Elements of Operator Algebras. In: Böttcher, A., Kaashoek, M.A., Lebre, A.B., dos Santos, A.F., Speck, FO. (eds) Singular Integral Operators, Factorization and Applications. Operator Theory: Advances and Applications, vol 142. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8007-7_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8007-7_12

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9401-2

  • Online ISBN: 978-3-0348-8007-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation