Operator space structure and amenability for Figa-Talamanca-Herz algebras

摘要

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' is an element of (1, infinity) with 1/p + 1/p' = 1, we use the operator space structure on CB(COL(L-p'(G))) to equip the Figa-Talamanca-Herz algebra A(p)(G) with an operator space structure, turning it into a quantized Banach algebra. Moreover, we show that, for p less than or equal to q less than or equal to 2 or 2 less than or equal to q less than or equal to p and amenable G, the canonical inclusion A(q) (G) subset of A(p) (G) is completely bounded (with cb-norm at most K-G(2), where K-G is Grothendieck's constant). As an application, we show that G is amenable if and only if A(p)(G) is operator amenable for all-and equivalently for one-p is an element of (1, infinity); this extends a theorem by Ruan. (C) 2003 Elsevier Inc. All rights reserved.