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F. We use the notation Ix I to denote the length of x. We remark that

Definition. Given a free subset F C G, a Haagerup function is a complex function u on G which is zero off [F] and such that: i) u(lG) = 1 and Iu(x) < l for all x; ii) u(x) = u(x )for x E F, iii) u(xy) = u(x)u(y) if Ixy = IxI + yI. It is not difficult to see that a Haagerup function may be defined by assigning arbitrary values of modulus not greater than one on F, and that the functions e-tIxl are exactly those Haagerup functions which have a constant real value on [F]. Theorem 1 below asserts that every Haagerup function is positive definite. Even though our work was inspired by [10] the methods of the present paper are different and simpler. Theorem 1 is used in Section 2 to study the Fourier-Stieltjes algebra B(G) of the free group. The key result is Theorem 2 which resembles, in many respects, Zygmund's theorem on Riesz products [17]. Generalizations of Reisz products and extensions of Zygmund's theorem have been extensively used to study the Fourier-Steiltjes algebra of an Abelian group by many authors [3, 11, 14, 15]. The Corollary proved in Section 3 implies the existence of unitary representations of free noncommutative groups, which are orthogonal, together with all their tensor