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On isomorphisms between weighted $L^p$-algebras

Part of: Abstract harmonic analysis

Published online by Cambridge University Press:  30 October 2020

Yulia Kuznetsova  and
Safoura Zadeh
Yulia Kuznetsova
Affiliation:
Laboratoire de Mathématiques de Besançon, Universite Bourgogne Franche-Comté, Besançon, Francee-mail:yulia.kuznetsova@univ-fcomte.fr
Safoura Zadeh*
Affiliation:
Laboratoire de Mathématiques de Besançon, Universite Bourgogne Franche-Comté, Besançon, France and Max-Planck-Institut für Mathematik, Bonn, Vivatsgasse 7, 53111, Germany
*
e-mail: jsafoora@gmail.com
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Abstract

Let G be a locally compact group and let $\omega $ be a continuous weight on G. In this paper, we first characterize bicontinuous biseparating algebra isomorphisms between weighted $L^p$ -algebras. As a result, we extend previous results of Edwards, Parrott, and Strichartz on algebra isomorphisms between $L^p$ -algebras to the setting of weighted $L^p$ -algebras. We then study the automorphisms of certain weighted $L^p$ -algebras on integers, applying known results on composition operators to classical function spaces.

Keywords

Locally compact groupsweighted Lp algebrastopological group isomorphismsbiseparating isomorphismsmall bound isomorphism

MSC classification

Primary: 43A20: $L^1$-algebras on groups, semigroups, etc.
Secondary: 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 853 - 866
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Y.K. is supported by the French “Investissements d'Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03). S.Z. is supported by the mobility program of the region of Bourgogne-Franche-Comté, France.

References

Aliprantis, C. D. and Burkinshaw, O., Positive operators . Pure and Applied Mathematics, 119, Academic Press, Inc., Orlando, FL, 1985.Google Scholar
Edwards, R. E., Bipositive and isometric isomorphisms of some convolution algebras . Canad. J. Math. 17(1965), 839–846. http://dx.doi.org/10.4153/CJM-1965-082-8 CrossRefGoogle Scholar
Fleming, R. J. and Jamison, J. E., Isometries on Banach spaces: function spaces . Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129, Chapman & Hall/CRC, Boca Raton, FL, 2003.Google Scholar
Gallardo-Gutiérrez, E. A. and Partington, J. R., Norms of composition operators on weighted Hardy spaces . Israel J. Math. 196(2013), 273–283. http://dx.doi.org/10.1007/s11856-012-0148-3 CrossRefGoogle Scholar
Gel’fand, I. M., Raĭkov, D. A., and Šilov, G. E., Kommutativnye normirovannye kol’tsa [Commutative normed rings] . Fiz.-Mat. Lit., Moscow, 1960.Google Scholar
Ghahramani, F. and Zadeh, S., Bipositive isomorphisms between Beurling algebras and between their second dual algebras . Canad. J. Math. 69(2017), 3–20. http://dx.doi.org/10.4153/CJM-2016-028-5 CrossRefGoogle Scholar
Gilbert, J. E., Interpolation between weighted ${L}^p$ -spaces. Ark. Mat. 10(1972), 235–249. http://dx.doi.org/10.1007/BF02384812 CrossRefGoogle Scholar
Kalton, N. J. and Wood, G. V., Homomorphisms of group algebras with norm less than $\sqrt{2}$ . Pacific J. Math. 62(1976), 439–460.CrossRefGoogle Scholar
Kawada, Y., On the group ring of a topological group . Math. Japon. 1(1948), 1–5.Google Scholar
Kuznetsova, Y., Invariant weighted algebras ${L}_p^w(G)$ . Math. Notes 84(2008), nos. 3–4, 529–537. http://dx.doi.org/10.1134/S0001434608090241 CrossRefGoogle Scholar
Kuznetsova, Y. and Molitor-Braun, C., Harmonic analysis of weighted ${L}^p$ -algebras. Expo. Math. 30(2012), 124–153. http://dx.doi.org/10.1016/j.exmath.2012.01.002 CrossRefGoogle Scholar
Kuznetsova, Y. N., Constructions of regular algebras ${L}_p^w(G)$ . Mat. Sbornik 200(2009), 75–88. http://dx.doi.org/10.1070/SM2009v200n02ABEH00993 Google Scholar
Lamperti, J., On the isometries of certain function-spaces . Pacific J. Math. 8(1958), 459–466.CrossRefGoogle Scholar
Lebedev, V. and Olevskii, A., ${C}^1$ changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers. Geom. Funct. Anal. 4(1994), 213–235. http://dx.doi.org/10.1007/BF01895838 CrossRefGoogle Scholar
Meyer-Nieberg, P., Banach lattices . Universitext, Springer-Verlag, Berlin, 1991. http://dx.doi.org/10.1007/978-3-642-76724-1 CrossRefGoogle Scholar
Nikolskiĭ, N. K., Spectral synthesis for the shift operator, and zeros in certain classes of analytic functions that are smooth up to the boundary . Dokl. Akad. Nauk SSSR 190(1970), 780–783.Google Scholar
Parrott, S. K., Isometric multipliers . Pacific J. Math. 25(1968), 159–166.CrossRefGoogle Scholar
Shapiro, J. H., Composition operators and classical function theory . Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. http://dx.doi.org/10.1007/978-1-4612-0887-7 CrossRefGoogle Scholar
Strichartz, S., Isomorphisms of group algebras . Proc. Amer. Math. Soc. 17(1966), 858–862. http://dx.doi.org/10.2307/2036268 CrossRefGoogle Scholar
Szehr, O. and Zarouf, R., A constructive approach to Schaeffer’s conjecture. Preprint, 2020. arXiv:1705.10704 Google Scholar
Wendel, J., On isometric isomorphism of group algebras . Pacific J. Math. 1(1951), 305–311.CrossRefGoogle Scholar
Wendel, J., Left centralizers and isomorphisms of group algebras . Pacific J. Math. 2(1952), 251–261.CrossRefGoogle Scholar
Wermer, J., On a class of normed rings . Ark. Mat. 2(1954), 537–551. http://dx.doi.org/10.1007/BF02591228 CrossRefGoogle Scholar
Wood, G. V., Isomorphisms of ${l}_p$ group algebras. Indiana Univ. Math. J. 50(2001), 1027–1045.CrossRefGoogle Scholar
Wood, G. V., Almost isometric ${}^{\ast }$ -homomorphisms of ${l}_p$ group algebras. In: Interaction between functional analysis, harmonic analysis, and probability (Columbia, MO, 1994), Lecture Notes in Pure and Appl. Math., 175, Dekker, New York, NY, 1996, pp. 461–466.Google Scholar
Zadeh, S., Isometric isomorphisms of Beurling algebras . J. Math. Anal. Appl. 1(2016), 1–13. http://dx.doi.org/10.1016/j.jmaa.2016.01.060 CrossRefGoogle Scholar
Zorboska, N., Composition operators on ${S}_a$ spaces. Indiana Univ. Math. J. 39(1990), 847–857. http://dx.doi.org/10.1512/iumj.1990.39.39041 CrossRefGoogle Scholar