Operator space structure and amenability for Figà-Talamanca–Herz algebras

Column and row operator spaces—which we denote by COL and ROW, respectively—over arbitrary Banach spaces were introduced by the first-named author; for Hilbert spaces, these definitions coincide with the usual ones. Given a locally compact group G and p,p′∈(1,∞) with 1p+1p′=1, we use the operator space structure on CB(COL(Lp′(G))) to equip the Figà-Talamanca–Herz algebra Ap(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p⩽q⩽2 or 2⩽q⩽p and amenable G, the canonical inclusion Aq(G)⊂Ap(G) is completely bounded (with cb-norm at most KG2, where KG is Grothendieckʹs constant). As an application, we show that G is amenable if and only if Ap(G) is operator amenable for all—and equivalently for one—p∈(1,∞); this extends a theorem by Ruan.