  For $1\leq p\leq \infty$ and $\alpha>0$, Besov spaces $B^p_\alpha$ play a key
role in the theory of $\alpha$-M\"obius invariant function spaces. In some
sense, $B^1_\alpha$ is the minimal $\alpha$-M\"obius invariant function space,
$B^2_\alpha$ is the unique $\alpha$-M\"obius invariant Hilbert space, and
$B^\infty_\alpha$ is the maximal $\alpha$-M\"obius invariant function space. In
this paper, under the $\alpha$-M\"obius invariant pairing and by the space
$B^\infty_\alpha$, we identify the predual and dual spaces of $B^1_\alpha$. In
particular, the corresponding identifications are isometric isomorphisms. The
duality theorem via the $\alpha$-M\"obius invariant pairing for $B^p_\alpha$
with $p>1$ is also given.
