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

Abstract

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 1 p + 1 p′ =1 , we use the operator space structure on CB( COL(L p′(G))) to equip the Figà-Talamanca–Herz algebra A p ( 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 A q ( G)⊂ A p ( G) is completely bounded (with cb-norm at most K G 2 , where K G is Grothendieck's constant). As an application, we show that G is amenable if and only if A p ( G) is operator amenable for all—and equivalently for one— p∈(1,∞); this extends a theorem by Ruan.