Operators on the Fourier Algebra with Weakly Compact Extensions

Introduction. We let G denote an infinite compact group, and G its dual. We use the notation of our book [3, Chapters 7 and 8]. Recall that -4(G) denotes the Fourier algebra of G (an algebra of continuous functions on G), and J2f°°(G) denotes its dual space under the pairing ( / , # ) , ( / G A (G), 0 Ç if°°(G)). Further, note if°°(G) is identified with the G*-algebra of bounded operators on L(G) commuting with left translation. The module action of A (G) on &(Ô) is defined by the following: for / Ç A (G), Ç if°°(G), / • * € i f œ « 5 ) by