Free products of completely positive maps and spectral sets

Abstract

Let [F, be the free group on n generators and for each word w E [F, let 1 WI be its length. Haagerup [lo] showed that for each 0 < r < 1, the function H,(wp) = r”” is a positive definite function on [F,,. Consider the positive definite function q,(n) = rinl on h. Then the free product function cpI * cpr on ff, = Z * H coincides with H, and the result was extended this way by de Michele and Fig&Talamanca [7] and by Boiejko [4, 51. In [S] it is proved that the free product of the unital positive defined functions ui: Gi + Z(Z) is still positive defined on the free product group * Gi. The correspondence between the positive defined functions on a discrete group G and the completely positive maps on the full C*-algebra C*(G) and the isomorphism C*(G, * G2) N C*(G,) Z C*(G,) suggested we consider amalgamated products of unital linear maps on the amalgamated product of a family of unital C*-algebras over a C*-subalgebra. In Section 3 we prove that the amalgamated product of a family of unital completely positive B-bimodule maps Qi: A, + C is completely positive on the “biggest” amalgamated free product i .A i. As an application of our main result, in Section 2 we obtain some results concerning the dilation of noncommutative families of operators.