Tensor products and multipliers of L1(G)-modules

AbstractIn [7] Rieffel is concerned with a locally compact group G and the problem of finding a concrete representation of Lp(G) ⊗L Lq(G), the tensor product of the L1(G)-modules Lp(G) and Lq(G). He shows that under some conditions Lp(G) ⊗L Lq(G) is isometrically isomorphic to a function space called Apq(G). Motivated by a theorem of Gaudry ([1]; Ch. V, Th. 2.5), we show that for compact Abelian G, p ⩾ 2, q ⩾ 2 this Apq(G) is simply the well-known space A(G) which consists of Fourier transforms of elements in L1(G) (Th. 4.8). Moreover we consider the tensor product L∞(G) ⊗L L∞(G), and show that the failure of the “multiplier result” ([3]; example 5) in L∞(G) ⊗L L∞(G) is general for non-discrete compact Abelian G (Th. 4.9)