  Motivated by recent experiments and band calculation results, we investigate
the possibility of superconductivity originating from the $a_{1g}$ band of
Na$_x$CoO$_2\cdot y$H$_2$O assuming the absence of $e_g'$ pockets. We adopt a
single band $U$-$V$ model that takes into account the local minimum of the
$a_{1g}$ band dispersion at the $\Gamma$ point. Using the fluctuation exchange
method, we find that an s-wave pairing with a gap that changes sign between the
inner and the outer Fermi surfaces is enhanced due to the coexistence of spin
fluctuations at wave vectors bridging the two Fermi surfaces and the charge
fluctuations near the K point. Possible relevance to the experimental results
is discussed.
