Operator space structure and amenability for Fig`a-Talamanca-Herz algebras.pdf

a r X i v : m a t h / 0 3 0 3 1 7 1 v 4 [ m a t h .F A ] 8 S e p 2 0 0 3 Operator space structure and amenability for Figa?-Talamanca–Herz algebras Anselm Lambert Matthias Neufang? Volker Runde? Abstract Column and row operator spaces — which we denote by COL and ROW, respec- tively — 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 K2 G , where KG is Grothendieck’s con- stant). 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. Keywords : operator spaces, operator sequence spaces, column and row spaces, locally compact groups, Figa?-Talamanca–Herz algebra, Fourier algebra, amenability, operator amenability. 2000 Mathematics Subject Classification: 43A15, 43A30, 46B70, 46J99, 46L07, 47L25 (primary), 47L50. Introduction The Fourier algebra A(G) of a locally compact group G was introduced by P. Eymard in [Eym 1]. If G is abelian with dual group Γ, then the Fourier transform induces an isometric isomorphism of A(G) and L1(Γ). Although the Fourier algebra is an invariant for G — like L1(G) —, its Banach algebraic amenability does not correspond well to the amenability of G — very much unlike L1(G): The group G is amenable if and only if L1(G) is amenable as a Banach algebra ([Joh 1]), but there are compact groups, among them SO(3), for which A(G) fails to be even weakly amenable ([Joh 2]). In fact, the only ?Part of the research for this paper was done whil