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 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 KG 2, 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 Ruan.

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Keywords Amenability, Column and row spaces, Figà-Talamanca-Herz algebra, Fourier algebra, Locally compact groups, Operator amenability, Operator sequence spaces, Operator spaces
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Journal Journal of Functional Analysis
Lambert, A. (Anselm), Neufang, T, & Runde, V. (Volker). (2004). Operator space structure and amenability for Figà-Talamanca-Herz algebras. Journal of Functional Analysis, 211(1), 245–269. doi:10.1016/j.jfa.2003.08.009