TENG-SUN LIU, ARNOUD VAN ROOIJ AND JU-KWEI1* Our notations are basically the same as those used in [3]. We use, however, C to denote the complex number field. Throughout the paper, G is a locally compact group with a left Haar measure λ. Instead of C Q0 (G), L P (G) etc. we write C oo , L P etc. We view L t as a subspace of M. We identify two functions that are equal almost everywhere.For a function f on G define /' bywhere Δ denotes the modular function of G. Then /" = / and (/•*)'= *'•/' for /^ei, If B is a left Banach module over L x (see [3; 32.14]), then J3* becomes a left Banach module by Every jeB induces the multiplier f\-*f*j.The following theorem is essentially due to Rieffel [6] and is proved in A Radon measure on G is a linear functional μ: C oo ~> C such that for every compact set CcG there exists a number c such that(J e C oo , Supp jcC).The Radon measures form a vector space which we denote by R. For μe R and for an f e L λ with compact support we define f*μeR and μ*feR by These formulas reduce to the familiar convolution formulas in case μeL p .Every Radon measure is a linear combination of positive Radon measures [1]. Thus, if μ e R and if X is a relatively compact Borel subset of G, we can in a natural way define μ(X). Further, if μe R and if A is any Borel set there is a unique ξ A μeR such that ξ A μ(X) = μ(X Π A) for all relatively compact Borel sets X. There ON SOME GROUP ALGEBRA MODULES 509 exists a unique |μ\ e R such that ξ A \μ\ = \ξ Λ μ| for every compact set A (see [1; Ch. 13]
G is a locally compact group with left Haar measure m. By C (G) we denote the Banach space of all bounded continuous ihnctions on G, while Co(G) + consists of all continuous functions G-~ [0, oc) that have compact support. A mean M on r or L~176 is cMled topologically left invariant (see [2; Def. 2.t.1]) if
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