Let $G$ be locally compact group. How we can show that
$$
M(C_0(G)\otimes C_b(G))=C_b(G,C_b(G)).
$$
($M(C_0(G)\otimes C_b(G))$ is the multiplier algebra of $C_0(G)\otimes C_b(G)$)

Which part can't you show? (Also, the fact $G$ is a group seems to play no role in what you want to prove.)
– Yemon ChoiMay 29 '13 at 1:18

I do not know how to define a bijection between them.
– zzzzMay 29 '13 at 6:37

I can define a map from $_M(C0(G) \otimes C_b(G))$ to $C_b(G,C_b(G))$ but I can not show that the map is sujective. by the way, you are right, I do not need to assume $G$ is a group.
– zzzzMay 29 '13 at 6:42

Try defining a map that goes the other way from right hand side to left hand side?
– Yemon ChoiMay 29 '13 at 16:02