Completely continuous elements of Banach algebras related to locally compact groups

Abstract

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.

The Acceptance of Mobile Health Services by Physicians: The Case of Iran

“…(iv) This follows from Theorem 2.1 and Corollary 2.3 of [16]. Now, we prove the main result of this section.…”

“…Then L ∞ (G) * and L ∞ 0 (G) * are Banach algebras with the first Arens product. One can prove that L ∞ (G) * and L ∞ 0 (G) * have right identities [5,9]; for more study see [1,2,[11][12][13][14]. Let M(G) be the measure algebra of G. Then M(G) with the convolution product is a unital Banach algebra and M(G) ∼ = C 0 (G) * , where C 0 (G) is the space of all complexvalued continuous functions on G that vanish at infinity [7].…”

Essential Character Amenability of Banach Algebras

“…This space was introduced and studied extensively by Lau and Pym [14]; see also [2,[15][16][17]. Now let G denote the dual group of G consisting of all continuous homomorphisms ρ from G into the circle group T, and define φ ρ ∈ σ(L 1 (G)) to be the character induced by ρ on L 1 (G); that is, φ ρ (a) = G ρ(x)a(x) dλ G (x) (a ∈ L 1 (G)).…”

Weakly compact multipliers on Banach algebras related to a locally compact group

“…The authors [14] have recently studied completely continuous elements of L ∞ 0 (G) * ; they have proved that the existence of a compact right multiplier on L ∞ 0 (G) * is equivalent to the compactness of G.…”