Intertwining Maps from Certain Group Algebras

In 17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin 19], we worked with general intertwining maps 3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem 4]. The present paper is a continuation of 17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of 26]. In 26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by 18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in 26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) 25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.