The existence of nonzero compact right multipliers and Arens regularity of weighted Banach algebras. (English) Zbl 1512.43002

Mehdipour, Mohammad Javad; Moghimi, Gholam Reza

Rocky Mt. J. Math. 52, No. 6, 2101-2112 (2022).

Summary: Let \(\omega\) be a weight function on a locally compact group \(\mathcal{G}\) and let \(M(\mathcal{G},\omega)_0^\ast\) be the subspace of \(M(\mathcal{G},\omega)^\ast\) consisting of all functionals that vanish at infinity. We first introduce an Arens product on \((M(\mathcal{G}, \omega)_0^\ast)^\ast\) under which it is a Banach algebra. We then show that the existence of a nonzero compact right multiplier on \((M(\mathcal{G}, \omega)_0^\ast)^\ast\) is equivalent to compactness of \(\mathcal{G}\). We also prove that if \((M(\mathcal{G}, \omega)_0^\ast)^\ast\) is Arens regular, then \(\mathcal{G}\) is discrete.

MSC:

43A10	Measure algebras on groups, semigroups, etc.
47B07	Linear operators defined by compactness properties
47B48	Linear operators on Banach algebras

Keywords:

locally compact group; weight function; measure algebra; compact right multiplier; Arens regularity

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