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Mat. Zametki, 2008, Volume 84, Issue 4, Pages 567–576 (Mi mz3866)  

This article is cited in 15 scientific papers (total in 15 papers)

Invariant Weighted Algebras $\mathscr L_p^w(G)$

Yu. N. Kuznetsova

All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences

Abstract: The paper is devoted to weighted spaces $\mathscr L_p^w(G)$ on a locally compact group $G$. If $w$ is a positive measurable function on $G$, then the space $\mathscr L_p^w(G)$, $p\ge1$, is defined by the relation $\mathscr L_p^w(G)=\{f:fw\in\mathscr L_p(G)\}$. The weights $w$ for which these spaces are algebras with respect to the ordinary convolution are treated. It is shown that, for $p>1$, every sigma-compact group admits a weight defining such an algebra. The following criterion is proved (which was known earlier for special cases only): a space $\mathscr L_1^w(G)$ is an algebra if and only if the function $w$ is semimultiplicative. It is proved that the invariance of the space $\mathscr L_p^w(G)$ with respect to translations is a sufficient condition for the existence of an approximate identity in the algebra $\mathscr L_p^w(G)$. It is also shown that, for a nondiscrete group $G$ and for $p>1$, no approximate identity of an invariant weighted algebra can be bounded.

Keywords: locally compact group, weighted space, weighted algebra, approximate identity, bounded approximate identity, $\sigma$-compact group, measurable function


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English version:
Mathematical Notes, 2008, 84:4, 529–537

Bibliographic databases:

UDC: 517.986
Received: 30.03.2007

Citation: Yu. N. Kuznetsova, “Invariant Weighted Algebras $\mathscr L_p^w(G)$”, Mat. Zametki, 84:4 (2008), 567–576; Math. Notes, 84:4 (2008), 529–537

Citation in format AMSBIB
\by Yu.~N.~Kuznetsova
\paper Invariant Weighted Algebras $\mathscr L_p^w(G)$
\jour Mat. Zametki
\yr 2008
\vol 84
\issue 4
\pages 567--576
\jour Math. Notes
\yr 2008
\vol 84
\issue 4
\pages 529--537

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    This publication is cited in the following articles:
    1. Kuznetsova Yu. N., “Example of a weighted algebra $\mathscr L^w_p(G)$ on an uncountable discrete group”, J. Math. Anal. Appl., 353:2 (2009), 660–665  crossref  mathscinet  zmath  isi  elib  scopus
    2. E. A. Gorin, “Regularity of group algebras”, Sb. Math., 200:8 (2009), 1165–1179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. Yu. N. Kuznetsova, “Constructions of regular algebras $\mathscr L_p^w(G)$”, Sb. Math., 200:2 (2009), 229–241  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. Abtahi F., “Lebesgue weighted $L^p$-algebra on locally compact groups”, Acta Math. Hungar., 133:4 (2011), 324–331  crossref  mathscinet  zmath  isi  elib  scopus
    5. Akbarbaglu I., Maghsoudi S., “On the generalized weighted Lebesgue spaces of locally compact groups”, Abstr. Appl. Anal., 2011 (2011), 947908, 15 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    6. Abtahi F., “Weighted $L^p$-spaces on locally compact groups”, Bull. Belg. Math. Soc. Simon Stevin, 19:2 (2012), 339–343  mathscinet  zmath  isi  elib
    7. Kuznetsova Yu., Molitor-Braun C., “Harmonic analysis of weighted $L^p$-algebras”, Expo. Math., 30:2 (2012), 124–153  crossref  mathscinet  zmath  isi  elib  scopus
    8. Abtahi F., “Weighted l-P-Space as a Segal Algebra”, Proc. Rom. Acad. Ser. A-Math. Phys., 14:2 (2013), 106–110  mathscinet  zmath  isi  elib
    9. Abtahi F., Amini H.G., Lotfi H.A., Rejali A., “Some Intersections of the Weighted l-P-Spaces”, Abstract Appl. Anal., 2013  crossref  mathscinet  isi  scopus
    10. Abtahi F., “Completely Continuous and Weakly Completely Continuous Abstract Segal Algebras”, Proc. Indian Acad. Sci.-Math. Sci., 123:4 (2013), 539–546  crossref  mathscinet  zmath  isi  scopus
    11. Abtahi F., “Pseudo-Contractibility of Weighted $L^p$-Algebras”, Analele Stiint. Univ. Ovidius C., 21:3 (2013), 5–15  mathscinet  zmath  isi
    12. Akbarbaglu I., Heydarpour M., Maghsoudi S., “Uniqueness of Uniform Norm and $C^*$-norm in $L_p(G,\omega)$”, Math. Slovaca, 64:2 (2014), 411–422  crossref  mathscinet  zmath  isi  elib  scopus
    13. Oztop S., Samei E., “Twisted Orlicz algebras, I”, Studia Math., 236:3 (2017), 271–296  crossref  mathscinet  zmath  isi  scopus
    14. Oztop S., Samei E., Shepelska V., “Weak Amenability of Weighted Orlicz Algebras”, Arch. Math., 110:4 (2018), 363–376  crossref  mathscinet  zmath  isi  scopus
    15. Oztop S., Samei E., Shepelska V., “Twisted Orlicz Algebras and Complete Isomorphism to Operator Algebras”, J. Math. Anal. Appl., 477:2 (2019), 1114–1132  crossref  isi
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