Operator space structure and amenability for Figà-Talamanca–Herz algebras
نویسندگان
چکیده
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 ′ (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 K G , 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.
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