Let G be a locally compact abelian group with a fixed Haar measure and let L1(G) be the group algebra of G equipped with the convolution product ? and the norm ?:?1. we denote by L1 0 (G) the suace of all functions f 2 L1(G), the usual Lebesgue space equipped with the essential supremum norm ?:?1, that for each ?, there is a compact subset K of G for which ?f?GnK?1 lt; ?, where ?GnK denotes the characteristic function G n K on G. It is well-known that the suace L1 0 (G) is a topologically introverted suace of L1(G), that is, for each n 2 L1 0 (G)? and f 2 L1 0 (G), the function nf 2 L1 0 (G), where ?nf; ?? = ?n; f??; ?f?; ? = ?f; ? ? ? for all ?; 2 L1(G). Hence L1 0 (G)? is a Banach algebra with the first Arens product ? defined by the formula ?m ? n; f? = ?m; nf? for all m; n 2 L1 0 (G)? and f 2 L1 0 (G). By rad(L1 0 (G)?) we denote the (Jacobson) radical of L1 0 (G)?. A linear map d : L1 0 (G)? ! L1 0 (G)? is called a derivation of L1 0 (G)? if d(m ? n) = d(m) ? n #43; m ? d(n) for all m; n 2 L1 0 (G)?. A linear map D : L1 0 (G)? ! L1 0 (G)? is called a generalized derivation of L1 0 (G)? if there exists a derivation d of L1 0 (G)? such that D(m ? n) = D(m) ? n #43; m ? d(n) for all m; n 2 L1 0 (G)?. The derivation d is called the associated derivation of D. In 1955 Singer and Wermer [19] initiated the study of range inclusion results for derivations on Banach algebras and proved that d(A) ? rad(A) when A is a commutative Banach algebra and d is continuous. In particular, d = 0 when A is semisimple. They also conjectured that the assumption of continuity is unnecessary. This became known as the Singer-Wermer conjecture and was finally proved in 1987 by Thomas [21]. Of course the same result does not hold in non-commutative Banach algebras, but there have been many non-commutative extensions of Singer-Wermer theorem in the literature [24, 25, 26, 14, 27, 11]. In [15] Posner gave a non-commutative version of the Singer-Wermer theorem for prime rings. He proved that the zero map is the only centralizing derivation on a non-commutative prime ring. In harmonic analysis, for a locally compact group G, the algebra L1 0 (G)?, equipped with the Arens product, is neither a commutative Banach algebra nor a prime ring when G is a non-discrete group. Thus, it is of interest to determine when a generalized derivation on L1 0 (G)? maps L1 0 (G)? into the radical.