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

Abstract

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\text{'}\in(1,\infty) with \frac{1}{p}+\frac{1}{p\text{'}}=1, we use the operator space structure on CB(COL(L^{p\text{'}}(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\leq q\leq 2 or 2\leq q\leq p and amenable G, the canonical inclusion A_{q}(G)\subset A_{p}(G) is completely bounded (with cb-norm at most K_{\mathbb{G}}^{2}, where K_{\mathbb{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\in(1,\infty); this extends a theorem by Z.-J. Ruan.