Abstract :
For a locally compact (non-compact) group G, consider L∞(G) as a left module over the algebra L1(G)∗∗, endowed with the first Arens product. We answer in the affirmative a question raised by Hofmeier–Wittstock (Math. Ann. 308 (1) (1997) 141) concerning the automatic boundedness of the corresponding module homomorphisms on L∞(G). In fact, we prove that an even stronger assertion holds true. Furthermore, we show that the theorem, first obtained by Ghahramani–McClure (Canad. Math. Bull. 35 (2) (1992) 180), on the automatic w∗-continuity of the (bounded) L∞(G)∗-module homomorphisms on L∞(G), can be sharpened in a similar fashion. To this end, a general factorization theorem for bounded families in L∞(G) is proved from which we shall derive both results.
