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
Let T be a locally compact group, Cl a measurable subset of Y, and let Ln denote the subspace of Ll(Y) consisting of all functions vanishing off fi. Assume that La is a subalgebra of L1^). We discuss the collection SRn(J0 of all module homomorphisms from La into an arbitrary Banach space K which is simultaneously a left LHr) module. We prove that na(K) = na(K0) © *a(.K*0s), where K0 is the collection of all k e K such that fk = 0, for all/eLl(r), and where K^bs consists of all elements of K which can be factored with respect to the module composition. We prove that 5"n(*'o) is the collection of linear continuous maps from La to K0 which are zero on a certain measurable subset of X. We reduce the determination of 9til(Ä_at,3) to the determination of Kr(Ä"alis). Denoting the topological conjugate space of K by K*, we prove that (A"ab3)* is isometrically isomorphic to SKn(.fv*). Finally, we discuss module homomorphisms R from Ln into L1(X) such that for each/e La, /(/vanishes off Y.
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