Skip to main content Accessibility help
×
×
Home

Injective Convolution Operators on ℓ(Γ) are Surjective

  • Yemon Choi (a1)
Abstract

Let Γ be a discrete group and let f ∈ ℓ1(Γ). We observe that if the natural convolution operator ρ f : ℓ(Γ) → ℓ(Γ) is injective, then f is invertible in ℓ1(Γ). Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra ℓ1(Γ).

We give simple examples to show that in general one cannot replace ℓ with ℓ p , 1 ≤ p < ∞, nor with L (G) for nondiscrete G. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on Γ, and give some partial results.

    • 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.

      Injective Convolution Operators on ℓ(Γ) are Surjective
      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.

      Injective Convolution Operators on ℓ(Γ) are Surjective
      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.

      Injective Convolution Operators on ℓ(Γ) are Surjective
      Available formats
      ×
Copyright
References
Hide All
[1] Deninger, C., Fuglede-Kadison determinants and entropy for actions of discrete amenable groups. J. Amer. Math. Soc. 19(2006), no. 3, 737758. doi:10.1090/S0894-0347-06-00519-4
[2] Deninger, C. and Schmidt, K., Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Preprint version, arXiv math.DS/0605723 v1.
[3] Deninger, C. and Schmidt, K., Expansive algebraic actions of discrete residually finite amenable groups and their entropy. Ergodic Theory Dynam. Systems 27(2007), no. 3, 769786. doi:10.1017/S0143385706000939
[4] Kadison, R. V. and Ringrose, J. R., Fundamentals of the theory of operator algebras. Vol. II. Pure and Applied Mathematics, 100, Academic Press Inc., Orlando, FL, 1986.
[5] Kaplansky, I., Fields and rings. The University of Chicago Press, Chicago, Ill.-London, 1969.
[6] Lam, T. Y., A first course in noncommutative rings. Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.
[7] Montgomery, M. S., Left and right inverses in group algebras. Bull. Amer. Math. Soc. 75(1969), 539540. doi:10.1090/S0002-9904-1969-12234-2
[8] Rudin, W., Functional analysis. Second ed. International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991.
[9] White, M. C., Characters on weighted amenable groups. Bull. London Math. Soc. 23(1991), no. 4, 375380. doi:10.1112/blms/23.4.375
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords

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