2001/11/30 by Volker Runde
Mathematics · #math.FA #math.OA #msc:43A15 #msc:43A30 #msc:46B70 #msc:46J99 #msc:46L07 #msc:47L25
published as Studia Math. 155 (2003), 152-170 · 20 pages; some typos eliminated
arxiv created 2003/01/02 · arxiv updated 2009/11/30
Let G be a locally compact group. We use the canonical operator space structure on the spaces Lp(G) for p ∈ [1,∞] introduced by G. Pisier to define operator space analogues OAp(G) of the classical Figa-Talamanca-Herz algebras Ap(G). If p ∈ (1,∞) is arbitrary, then Ap(G) ⊂ OAp(G) such that the inclusion is a contraction; if p = 2, then OA2(G) ≅ A(G) as Banachspaces spaces, but not necessarily as operator spaces. We show that OAp(G) is a completely contractive Banach algebra for each p ∈ (1,∞), and that OAq(G) ⊂ OAp(G) completely contractively for amenable G if 1 < p ≤ q ≤ 2 or 2 ≤ q ≤ p < ∞. Finally, we characterize the amenability of G through the existence of a bounded approximate identity in OAp(G) for one (or equivalently for all) p ∈ (1,∞).