  It is well known that the discretization of fractional diffusion equations
(FDEs) with fractional derivatives $\alpha\in(1,2)$, using the so-called
weighted and shifted Gr\"unwald formula, leads to linear systems whose
coefficient matrices show a Toeplitz-like structure. More precisely, in the
case of variable coefficients, the related matrix sequences belong to the
so-called Generalized Locally Toeplitz (GLT) class. Conversely, when the given
FDE have constant coefficients, using a suitable discretization, we encounter a
Toeplitz structure associated to a nonnegative function $\mathcal{F}_\alpha$,
called the spectral symbol, having a unique zero at zero of real positive order
between one and two. For the fast solution of such systems by preconditioned
Krylov methods, several preconditioning techniques have been proposed in both
the one and two dimensional cases. In this note we propose a new preconditioner
denoted by $\mathcal{P}_{\mathcal{F}_\alpha}$ which belongs to the $\tau$
algebra and it is based on the spectral symbol $\mathcal{F}_\alpha$. Comparing
with some of the previously proposed preconditioners, we show that although the
low band structure preserving preconditioners are more effective in the
one-dimensional case, the new preconditioner performs better in the more
challenging multi-dimensional setting.
