Let G1, G2 be locally compact groups. We prove in this paper that if T is an isometric isomorphism from the Banach algebra LUC(G1) * (the continuous dual of the Banach space of left uniformly continuous functions on G1, equipped with Arens multiplication) onto LUC(G2)*I then T maps M(G1) onto M(G2) and LI(G,) onto L'(G2). We also prove that any isometric isomorphism from LI(G,)** (second conjugate algebra of L' (GI)) onto Ll (G2) maps L' (G1) onto L' (G2) . 0. INTRODUCTION AND PRELIMINARIES Let G1, G2 be locally compact groups. Let M(Gi), i = 1, 2, be the Banach algebra of regular Borel measures on Gi . A well-known result of B. E. Johnson [10] asserts that if T is an isometric isomorphism from M(G1) onto M(G2), then T maps Ll (G,) onto L (G2) (and hence G1 and G2 must be isomorphic by Wendel's theorem [21]). In this paper we prove (Theorem 3.1 (c)), among other things, that if T is an isometric isomorphism from Ll(G1)** onto L1(G2)**, then T maps L1(G1) onto Ll (G2). This answers affirmatively a question raised in [4]. Theorem 3.1 (c) was proved for abelian locally compact groups by Lau and Losert in 13], and for compact and discrete groups by Ghahramani and Lau in [4]. Let G be a locally compact group. Let C(G) denote the space of bounded continuous complex-valued functions on G with the sup norm topology, and LUC(G) denote the closed subspace of bounded left uniformly continuous functions on G, i.e. all f E C(G) such that the map x -?lxf from G into C(G) is continuous, where (lxf)(y) = f(xy), x, y E G. Then LUC(G)* is a Banach algebra with the Arens multiplication defined by (nm, f) = (n, m1f), n, m E LUC(G)*, f E LUC(G), where mlf(x) = (m, lxf) x E G. Furthermore, M(G) may be identified with a closed subspace of LUC(G)* by the natural embedding (,u, f) = f f(x) du(x), f E LUC(G), ,u E M(G) . It was Received by the editors November 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 43A20.