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

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

DOI: https://doi.org/10.4213/mzm3866

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

Bibliographic databases:

UDC: 517.986

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
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• http://mi.mathnet.ru/eng/mz3866
• https://doi.org/10.4213/mzm3866
• http://mi.mathnet.ru/eng/mz/v84/i4/p567

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. E. A. Gorin, “Regularity of group algebras”, Sb. Math., 200:8 (2009), 1165–1179
3. Yu. N. Kuznetsova, “Constructions of regular algebras $\mathscr L_p^w(G)$”, Sb. Math., 200:2 (2009), 229–241
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