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We show that the (semi-infinite) Ablowitz-Ladik (AL) hierarchy admits acenterless Virasoro algebra of master symmetries in the sense of Fuchssteiner[Progr. Theoret. Phys. 70 (1983), 1508-1522]. An explicit expression for thesesymmetries is given in terms of a slight generalization of the Cantero, Moraland Vel\'azquez (CMV) matrices [Linear Algebra Appl. 362 (2003), 29-56] andtheir action on the tau-functions of the hierarchy is described. The use of theCMV matrices turns out to be crucial for obtaining a Lax pair representation ofthe master symmetries. The AL hierarchy seems to be the first example of anintegrable hierarchy which admits a full centerless Virasoro algebra of mastersymmetries, in contrast with the Toda lattice and Korteweg-de Vries hierarchieswhich possess only "half of" a Virasoro algebra of master symmetries, asexplained 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].

A Centerless Virasoro Algebra of Master Symmetries for the Ablowitz-Ladik Hierarchy