  Let $\omega $ be a weight function on a locally compact group G mand let $
M_* (G, \omega ) $ be the subspace of $ M(G , \omega )^* $ consisting of all
functionals that vanish at infinity. In this paper, we first investigate the
Arens regularity of $ M_* (G, \omega )^* $ and show that $ M_* (G, \omega )^* $
is Arnes regular if and only if G is finite or $ \omega $ is zero cluster. This
result is an answer to the question posed and it improves some well-known
results. We also give necessary and sufficient criteria for the weight function
spaces $ Wap(G , 1/ \omega ) $ and $ Wap(G , 1/ \omega ) $ to be equal to $ C_b
(G , 1/ \omega ) $. We prove that for non-compact group G, the Banach algebra $
M_* (G, \omega )^* $ is Arnes regular if and only if $ Wap(G , 1/ \omega ) =
C_b (G , 1/ \omega ) $. We then investigate amenability of $ M_* (G, \omega )^*
$ and prove that $ M_* (G, \omega )^* $ is amenable and Arnes regular if and
only if G is finite.
