  We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits a
centerless Virasoro algebra of master symmetries in the sense of Fuchssteiner
[Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for these
symmetries is given in terms of a slight generalization of the Cantero, Moral
and Vel\'azquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] and
their action on the tau-functions of the hierarchy is described. The use of the
CMV matrices turns out to be crucial for obtaining a Lax pair representation of
the master symmetries. The AL hierarchy seems to be the first example of an
integrable hierarchy which admits a full centerless Virasoro algebra of master
symmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchies
which possess only "half of" a Virasoro algebra of master symmetries, as
explained in Adler and van Moerbeke [Duke Math. J. 80 (1995), 863-911],
Damianou [Lett. Math. Phys. 20 (1990), 101-112] and Magri and Zubelli [Comm.
Math. Phys. 141 (1991), 329-351].
