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The present paper began as a natural outgrowth of our first paper, where we characterized the module homomorphisms from group algebras into a fairly restrictive class of group algebra modules. We now investigate module homomorphisms from group algebras into a more general class of group algebra modules. Although the two papers are thus related, they can be read quite independently.Section 2 contains our extension, Theorem 2.1, of P. J. Cohen's theorem on factorization in Banach algebras (1). Our extension is to Banach modules over Banach algebras equipped with an approximate identity. We should mention first that J.-K. Wang observed the existence of such a generalization, and secondly, that our proof requires no ideas different from those in Cohen's proof. Nevertheless, we include a proof that condenses the original proof considerably.

Some time ago, J. G. Wendel proved that the operators on the group algebra L1(G) which commute with convolution correspond in a natural way to the measure algebra M(G) (13). One might ask if Wendel's theorem can be restated in a more general setting. It is this question that is the point of departure for our present paper. Let K be a Banach module over L1(G). Our interest is in operators from L1(G) into K, and from K into L∞(G), which commute with the module composition (where L∞(G) is thought of as a module over L1(G) also). Such operators we call (L1(G), K)- and (K, L∞(G))-homomorphisms, respectively. Investigations of various other kinds of module homomorphisms occur in A. Figà-Talamanca (6) and B. E. Johnson (9; 10).

Abstract. Let F be a locally compact group and K a Banach space. The left L1^) module K is by definition absolutely continuous under the composition * if for k e K there exist f e LX{T), k' e Kviith k =f * k'. If the locally compact Hausdorff space X is a transformation group over Y and has a measure quasi-invariant with respect to T, then L*(X) is an absolutely continuous L1(r) module-the main object we study. If Y<£X is measurable, let LY consist of all functions in L1(X) vanishing outside Y. For OsT not locally null and B a closed linear subs pace of K, we observe the connection between the closed linear span (denoted La * B) of the elements/* k, with fe La and k e B, and the collection of functions of B shifted by elements in Ci. As a result, a closed linear subspace of L1(Y) is an Lz for some measurable Z<=X if and only if it is closed under pointwise multiplication by elements of L"(X). This allows the theorem stating that if Clç^T and KÇJTare both measurable, then there is a measurable subset Z of X such that La * LY = LZ. Under certain restrictions on T, we show that this Z is essentially open in the (usually stronger) orbit topology on X. Finally we prove that if Ci and Y are both relatively sigma-compact, and if also La * Ly^Ly, then there exist Hi and Yx locally almost everywhere equal to Ci and Y respectively, such that Cl¡ Yx £ Yx ; in addition we characterize those Cl and Y for which La * La = La and La* LY = LY.1. Introduction. This paper, and the one which follows, arise quite naturally from our earlier papers [3] and [4]. Let us see how. Take T as a locally compact group, and L^T) the Banach space of integrable functions on I\ If we let K be an arbitrary left Fx(r) module, we may inquire what are the left module homomorphisms from L^T) to K. In [3], amongst other things, we give a (not quite complete) solution to the general question, and then give complete solutions in case K=LP(T), pe [\, co]. In [4] we assume that F acts on a given locally compact space A' as a transformation group and that mx is a measure on X quasi-invariant with respect to T. Then we show that L"(X) may be rendered as a left Fx(r) module, to which we may ask what are the left module homomorphisms from L\T) to LP(X).The present investigations start at that point. In this paper we discuss the more general aspects of Banach spaces K which can be represented as left L1(T) modules. We denote the module composition by *. We pay particular attention to those modules whose elements are factorable (i.e., keK implies that there is an