Skip to main content Accessibility help
×
Home

Topologically simple Banach algebras with derivation

  • El Hossein Illoussamen (a1) and Volker Runde (a2)

Abstract

It is not known if a commutative, topologically simple, radical Banach algebra exists. If, however, every derivation on such an algebra is continuous, this yields the automatic continuity of all derivations on commutative, semiprime Banach algebras. Utilising techniques used by Thomas in his proof of the Singer-Wermer conjecture, we show that, if A is a commutative, topologically simple Banach algebra with a non-zero derivation on it, then a quotient of a certain localisation of A has a power series structure. A pivotal role is played by what we call ample sets of denominators.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Topologically simple Banach algebras with derivation
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Topologically simple Banach algebras with derivation
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Topologically simple Banach algebras with derivation
      Available formats
      ×

Copyright

References

Hide All
[1]Allan, G. R., ‘Embedding the algebra of formal power series in a Banach algebra’, Proc. London Math. Soc. (3) 25 (1972), 329340.
[2]Allan, G.R. Elements of finite closed descent in a Banach algebra, J. London Math. Soc. (2) 7 (1973), 462466.
[3]Atiyah, M.F. and MacDonald, I. G., Introduction to commutative algebra (Addison-Wesley, Massachusetts, London, Ontario, 1969).
[4]Curtis, P.C. Jr, ‘Derivations in commutative Banach algebras’, in Radical Banach algebras and automatic continuity, (Bachar, J.M. et al. , Editors), Springer Lecture Notes in Mathematics 975 (Springer Verlag, Berlin, Heidelberg, New York, 1983), pp. 328333.
[5]Cusack, J., ‘Automatic continuity and topologically simple radical Banach algebras’, J. London Math. Soc. (2) 16 (1977), 493500.
[6]Dixon, P.G., ‘Semiprime Banach algebras’, J. London Math. Soc. (2) 6 (1973), 676678.
[7]Esterle, J., ‘Mittag-Leffler methods in the theory of Banach algebras and a new approach to Michael's problem’, in Proceedings of the Conference on Banach Algebras and Several Complex Variables, (Greenleaf, F. and Gulick, D., Editors), Contemporary Mathematics 32 (Amer. Math. Soc, Providence R.I., 1984), pp. 107129.
[8]Garimella, R.V., ‘Continuity of derivations on some semiprime Banach algebras’, Proc. Amer. Math. Soc. 99 (1987), 289292.
[9]Illoussamen, E., ‘Continuité des dérivations et des épimorphismes dans certaines algèbres de Banach’, Rend. Circ. Mat. Paĺermo (2) 44 (1995), 173186.
[10]Johnson, B.E., ‘Continuity of derivations on commutative algebras’, Amer. J. Math. 91 (1969), 110.
[11]Runde, V., ‘Automatic continuity of derivations and epimorphisms’, Pacific J. Math. 147 (1991), 365374.
[12]Singer, I.M. and Wermer, J., ‘Derivations on commutative normed algebras’, Math. Ann. 129 (1955), 260264.
[13]Suciu, I., ‘Eine natürliche Erweiterung der kommutativen Banachalgebren’, Rev. Roum. Math. Pures Appl. 7 (1962), 483491.
[14]Suciu, I., ‘Bruchalgebren der Banachalgebren’, Rev. Roum. Math. Pures Appl. 8 (1963), 313316.
[15]Thomas, M.P., ‘The image of a derivation is contained in the radical’, Ann. of Math. 128 (1988), 435460.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed