Compact Left Multipliers on Banach Algebras Related to Locally Compact Groups

Abstract

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.

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

“…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].…”

“…Remark 2.5 In Theorems 2.2, 2.4 and Corollary 2.3, we consider a Banach algebra A such that either eA is semisimple when A has a right identity e or A/ran(A) is semisimple. ( ) Let us remark that if G is a locally compact group, then the Banach algebras L 1 (G), M(G) and L ∞ 0 (G) * satisfy the condition ( ); For an extensive study of these Banach algebras see [8,9,10,11]. Also, C * -algebras, semisimple Banach algebras with a bounded approximate identity and Banach algebras with rad(A) = ran(A) satisfy the condition ( ).…”

The Strong Dual of Measure Algebras With Certain Locally Convex Topologies

“…Alongside this work, we have studied various aspects of analysis on GL 0 (G) and its dual as well, which we shall publish elsewhere. It should be mentioned that some structures similar to GL 0 (G) in the study of Banach algebras related to G can be seen in several recent works; see, for example, [9,11,15].…”

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)).…”

“…for all F ∈ L ∞ 0 (G) * , where a 0 ∈ L 1 (G) with φ ρ (a 0 ) = 1. Note that L ∞ 0 (G) * has a bounded approximate identity if and only if G is discrete; see [17,Proposition 3.1]. P 5.3.…”