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

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 $p, p^{'} \in (1, \infty)$ with $\frac{1}{p} + \frac{1}{p’} = 1 $, w e u s e t h e o p e r a t o r s p a c e s t r u c t u r e o n$ 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 \leq q \leq 2 $o r$ 2 \leq q \leq p $a n d a m e n a b l e$ G $, t h e c a n o n i c a l i n c l u s i o n$ A_q(G) \subset A_p(G) $i s c o m p l e t e l y b o u n d e d (w i t h c b - n o r m a t m o s t$ K_G^2 $, w h e r e$ K_G $i s G r o t h e n d i e c k^{'} s c o n s t a n t) . A s a n a p p l i c a t i o n, w e s h o w t h a t$ 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.

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 $p, p^{'} \in (1, \infty)$ with $\frac{1}{p} + \frac{1}{p’} = 1 $, w e u s e t h e o p e r a t o r s p a c e s t r u c t u r e o n$ 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 \leq q \leq 2 $o r$ 2 \leq q \leq p $a n d a m e n a b l e$ G $, t h e c a n o n i c a l i n c l u s i o n$ A_q(G) \subset A_p(G) $i s c o m p l e t e l y b o u n d e d (w i t h c b - n o r m a t m o s t$ K_G^2 $, w h e r e$ K_G $i s G r o t h e n d i e c k^{'} s c o n s t a n t) . A s a n a p p l i c a t i o n, w e s h o w t h a t$ 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.

文献摘要