Solution to a conjecture by Hofmeier-Wittstock

摘要

For a locally compact (non-compact) group G, consider L-infinity(G) as a left module over the algebra Li(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-infinity(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-infinity(G)*-rnodule homomorphisms on can be sharpened in a similar fashion. To this end, a general factorization theorem for bounded families in L-infinity(G) is proved from which we shall derive both results. (C) 2004 Elsevier Inc. All rights reserved.