Let G be a locally compact group and be the Banach space of all essentially bounded measurable functions on G vansihing an infinity. Here, we study some families of right completely continuous elements in the Banach algebra equipped with an Arens type product. As the main result, we show that has a certain right completely continuous element if and only if G is compact.

We deal with the dual Banach algebras L ∞ 0 (G) * for a locally compact group G. We investigate compact left multipliers on L ∞ 0 (G) * , and prove that the existence of a compact left multiplier on L ∞ 0 (G) * is equivalent to compactness of G. We also describe some classes of left completely continuous elements in L ∞ 0 (G) * .2000 Mathematics subject classification: primary 43A15, 46H05, 47B48; secondary 43A20, 47B07.

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