  We prove general uniqueness results for radial solutions of linear and
nonlinear equations involving the fractional Laplacian $(-\Delta)^s$ with $s
\in (0,1)$ for any space dimensions $N \geq 1$. By extending a monotonicity
formula found by Cabre and Sire \cite{CaSi-10}, we show that the linear
equation $(-\Delta)^s u+ Vu = 0$ in $\mathbb{R}^N$ has at most one radial and
bounded solution vanishing at infinity, provided that the potential $V$ is a
radial and non-decreasing. In particular, this result implies that all radial
eigenvalues of the corresponding fractional Schr\"odinger operator
$H=(-\Delta)^s + V$ are simple. Furthermore, by combining these findings on
linear equations with topological bounds for a related problem on the upper
half-space $\mathbb{R}^{N+1}_+$, we show uniqueness and nondegeneracy of ground
state solutions for the nonlinear equation $(-\Delta)^s Q + Q - |Q|^{\alpha} Q
= 0$ in $\mathbb{R}^N$ for arbitrary space dimensions $N \geq 1$ and all
admissible exponents $\alpha >0$. This generalizes the nondegeneracy and
uniqueness result for dimension N=1 recently obtained by the first two authors
in \cite{FrLe-10} and, in particular, the uniqueness result for solitary waves
of the Benjamin--Ono equation found by Amick and Toland \cite{AmTo-91}.
