Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:20:05.410Z Has data issue: false hasContentIssue false
  • English
  • Français

Injective Convolution Operators on ℓ∞(Γ) are Surjective

Published online by Cambridge University Press:  20 November 2018

Yemon Choi
Yemon Choi*
Affiliation:
Department of Mathematics, University of Manitoba, Winnipeg, MB, andDepartment of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N 5E6 e-mail: y.choi.97@cantab.net
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $\Gamma $ be a discrete group and let $f\,\in \,{{l}^{1}}(\Gamma )$. We observe that if the natural convolution operator $\rho f\,:\,{{l}^{\infty }}(\Gamma )\,\to \,{{l}^{\infty }}(\Gamma )$ is injective, then $f$ is invertible in ${{l}^{1}}(\Gamma )$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra ${{l}^{1}}(\Gamma )$.

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

Keywords

43A2046L0543A22
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 53 , Issue 3 , 01 September 2010 , pp. 447 - 452
Copyright
Copyright © Canadian Mathematical Society 2010

References

[1] Deninger, C., Fuglede-Kadison determinants and entropy for actions of discrete amenable groups. J. Amer. Math. Soc. 19(2006), no. 3, 737–758. doi:10.1090/S0894-0347-06-00519-4Google Scholar
[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.Google Scholar
[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, 769–786. doi:10.1017/S0143385706000939Google Scholar
[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.Google Scholar
[5] Kaplansky, I., Fields and rings. The University of Chicago Press, Chicago, Ill.-London, 1969.Google Scholar
[6] Lam, T. Y., A first course in noncommutative rings. Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991.Google Scholar
[7] Montgomery, M. S., Left and right inverses in group algebras. Bull. Amer. Math. Soc. 75(1969), 539–540. doi:10.1090/S0002-9904-1969-12234-2Google Scholar
[8] Rudin, W., Functional analysis. Second ed. International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991.Google Scholar
[9] White, M. C., Characters on weighted amenable groups. Bull. London Math. Soc. 23(1991), no. 4, 375–380. doi:10.1112/blms/23.4.375Google Scholar