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Extensions of Positive Definite Functions on Amenable Groups

  • M. Bakonyi (a1) and D. Timotin (a2)

Abstract

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.

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References

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[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, 14011406. doi:10.1090/S0002-9939-01-06288-8
[2] Bakonyi, M. and Nævdal, G., The finite subsets of 2 having the extension property. J. London Math. Soc. 62(2000), no. 3, 904916. doi:10.1112/S0024610700001496
[3] Bakonyi, M. and Timotin, D., Extensions of positive definite functions on free groups. J. Funct. Anal. 246(2007), no. 1, 3149. doi:10.1016/j.jfa.2007.01.015
[4] Dixmier, J., C*-algebras. North-Holland Mathematical Library, 15, North Holland, Amsterdam-New York, 1977.
[5] Exel, R., Hankel matrices over right ordered amenable groups. Canad. Math. Bull. 33(1990), no. 4, 404415.
[6] Gabardo, J.-P., Trigonometric moment problems for arbitrary finite subsets of n . Trans. Amer. Math. Soc., 350(1998), no. 11, 44734498. doi:10.1090/S0002-9947-98-02091-1
[7] Golumbic, M. C., Algorithmic graph theory and perfect graphs. Academic Press, New York, 1980.
[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), 109125. doi:10.1016/0024-3795(84)90207-6
[9] Rudin, W., The extension problem for positive-definite functions. Illinois J. Math. 7(1963), 532539.
[10] Timotin, D., Completions of matrices and the commutant lifting theorem. J. Funct. Anal. 104(1992), no. 2, 291298. doi:10.1016/0022-1236(92)90002-Z
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Extensions of Positive Definite Functions on Amenable Groups

  • M. Bakonyi (a1) and D. Timotin (a2)

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