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

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.