Isometric isomorphisms between Banach algebras related to locally compact groups

Abstract

Let G 1 {G_1} , G 2 {G_2} be locally compact groups. We prove in this paper that if T T is an isometric isomorphism from the Banach algebra LUC ( G 1 ) ∗ {\text {LUC}}{({G_1})^*} (the continuous dual of the Banach space of left uniformly continuous functions on G 1 {G_1} , equipped with Arens multiplication) onto LUC ( G 2 ) ∗ {\text {LUC}}{({G_2})^*} , then T T maps M ( G 1 ) M({G_1}) onto M ( G 2 ) M({G_2}) and L 1 ( G 1 ) {L^1}({G_1}) onto L 1 ( G 2 ) {L^1}({G_2}) . We also prove that any isometric isomorphism from L 1 ( G 1 ) ∗ ∗ {L^1}{({G_1})^{**}} (second conjugate algebra of L 1 ( G 1 ) {L^1}({G_1}) ) onto L 1 ( G 2 ) ∗ ∗ {L^1}{({G_2})^{**}} maps L 1 ( G 1 ) {L^1}({G_1}) onto L 1 ( G 2 ) {L^1}({G_2}) .