2003/03/31 by Anselm Lambert, Matthias Neufang, Volker Runde
Mathematics · #math.FA #math.OA #msc:43A15 #msc:43A30 #msc:46B70 #msc:46J99 #msc:46L07 #msc:47L25 #msc:47L50
published as J. Funct. Anal. 211 (2004), 245-269 · 25 pages; some minor, hopefully clarifying revisions
arxiv created 2003/09/08 · arxiv updated 2009/11/30
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.