Hostname: page-component-6bb9c88b65-zjgpb Total loading time: 0 Render date: 2025-07-27T16:24:47.345Z Has data issue: false hasContentIssue false
  • English
  • Français

Extensions of Positive Definite Functions on Amenable Groups

Published online by Cambridge University Press:  20 November 2018

M. Bakonyi  and
D. Timotin
M. Bakonyi
Affiliation:
Department of Mathematics and Statistics, Georgia State University, Atlanta, GA, U.S.A.e-mail: mbakonyi@gsu.edu
D. Timotin
Affiliation:
Institute of Mathematics of the Romanian Academy, Bucharest, Romaniae-mail: Dan.Timotin@imar.ro
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 $S$ be a subset of an amenable group $G$ such that $e\,\in \,S$ and ${{S}^{-1}}\,=\,S$. The main result of this paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as corollaries. New applications are also presented.

Keywords

43A3547A5720E05

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 1 , 01 March 2011 , pp. 3 - 11
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Bakonyi, M., The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group. Proc. Amer. Math. Soc. 130(2002), no. 5, 1401–1406. doi:10.1090/S0002-9939-01-06288-8Google Scholar
[2] Bakonyi, M. and Nævdal, G., The finite subsets of ℤ 2 having the extension property. J. London Math. Soc. 62(2000), no. 3, 904–916. doi:10.1112/S0024610700001496Google Scholar
[3] Bakonyi, M. and Timotin, D., Extensions of positive definite functions on free groups. J. Funct. Anal. 246(2007), no. 1, 31–49. doi:10.1016/j.jfa.2007.01.015Google Scholar
[4] Dixmier, J., C*-algebras. North-Holland Mathematical Library, 15, North Holland, Amsterdam-New York, 1977.Google Scholar
[5] Exel, R., Hankel matrices over right ordered amenable groups. Canad. Math. Bull. 33(1990), no. 4, 404–415.Google Scholar
[6] Gabardo, J.-P., Trigonometric moment problems for arbitrary finite subsets of ℤ n . Trans. Amer. Math. Soc., 350(1998), no. 11, 4473–4498. doi:10.1090/S0002-9947-98-02091-1Google Scholar
[7] Golumbic, M. C., Algorithmic graph theory and perfect graphs. Academic Press, New York, 1980.Google Scholar
[8] Grone, R., Johnson, C. R., de Sá, E. M., and Wolkowicz, H., Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58(1984), 109–125. doi:10.1016/0024-3795(84)90207-6Google Scholar
[9] Rudin, W., The extension problem for positive-definite functions. Illinois J. Math. 7(1963), 532–539.Google Scholar
[10] Timotin, D., Completions of matrices and the commutant lifting theorem. J. Funct. Anal. 104(1992), no. 2, 291–298. doi:10.1016/0022-1236(92)90002-ZGoogle Scholar