Intertwining Maps from Certain Group Algebras

In [ 17 , 18 , 19 ], we began to investigate the continuity properties of homomorphisms from (non‐abelian) group algebras. Already in [ 19 ], we worked with general intertwining maps [ 3 , 12 ]. These maps not only provide a unified approach to both homomorphisms and derivations, but also have some significance in their own right in connection with the cohomology comparison problem [ 4 ]. The present paper is a continuation of [ 17 , 18 , 19 ]; this time we focus on groups which are connected or factorizable in the sense of [ 26 ]. In [ 26 ], G. A. Willis showed that if G is a connected or factorizable, locally compact group, then every derivation from L 1 ( G ) into a Banach L 1 ( G )‐module is automatically continuous. For general intertwining maps from L 1 ( G ), this conclusion is false: if G is connected and, for some n ∈N, has an infinite number of inequivalent, n ‐dimensional, irreducible unitary representations, then there is a discontinuous homomorphism from L 1 ( G into a Banach algebra by [ 18 , Theorem 2.2] (provided that the continuum hypothesis is assumed). Hence, for an arbitrary intertwining map θ from L 1 ( G ), the best we can reasonably hope for is a result asserting the continuity of θ on a ‘large’, preferably dense subspace of L 1 ( G ). Even if the target space of θ is a Banach module (which implies that the continuity ideal I (θ) of θ is closed), it is not a priori evident that θ is automatically continuous: the proofs of the automatic continuity theorems in [ 26 ] rely on the fact that we can always confine ourselves to restrictions to L 1 ( G ) of derivations from M ( G ) [ 25 , Lemmas 3.1 and 3.4]. It is not clear if this strategy still works for an arbitrary intertwining map from L 1 ( G ) into a Banach L 1 ( G )‐module.