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

الأصل · EN

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(Lᵖ'(G))) to equip the Figa-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 Grothendieck's 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 Z.-J. Ruan.