Abstract

Let $G_1$ and $G_2$ be LCA groups with identities being $e_1$ and $e_2$, and let $\omega$ be a (Borel measurable) weight function on $G_1 \times G_2$. Let $\overline{\omega}(s, t) = \omega(s, e_2)\omega(e_1, t) \; ((s, t) \in G_1 \times G_2)$. Then $\overline{\omega}$ is also a weight function on $G_1 \times G_2$. In this small note, it is proved that the Beurling algebra $L^1(G_1 \times G_2, \omega)$ has unique uniform norm property iff $L^1(G_1 \times G_2, \overline{\omega})$ has the same property. This result is important because the above statement does not hold true for some properties.

Keywords and Phrases

LCA Group, Weight, Beurling Algebra, and Unique Uniform Norm Property (UUNP).

A.M.S. subject classification

Primary 46J05, Secondary 46J10.

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