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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.
UDC:
517.986 Received: 30.03.2007
DOI:
10.4213/mzm3866
Fulltext:
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