INTERTWINING MAPS FROM CERTAIN GROUP ALGEBRAS

Abstract

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 L¹(G) into a Banach L¹(G)-module is automatically continuous. For general intertwining maps from L¹(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 L¹(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 L¹(G), the best we can reasonably hope for is a result asserting the continuity of θ on a ‘large’, preferably dense subspace of L¹(G). Even if the target space of θ is a Banach module (which implies that the continuity ideal [Iscr ](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 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 arbitrary intertwining map from L¹(G) into a Banach LL¹(G)-module.