Extensions of operator valued positive definite functions and commutant lifting on ordered groups

Let Omega be a locally compact abelian ordered group. We say that Omega has the extension property if every operator valued continuous positive definite function on an interval of Omega has a positive definite extension to the whole group and we say that Omega has the commutant lifting property if a natural extension of the commutant lifting theorem holds on Omega. We give a characterization of the groups having the extension Property in terms of unitary extensions of a particular class of multiplicative family of partial isometrics. It is proved that ifa group hits the extension property and satisfies an archimedean condition then it has the commutant lifting property. It is also proved that if the ordered group Gamma has the extension property and satisfies an archimedean condition then Omega = Gamma x Z with the lexicographic order has the extension property. As an application we obtain that the groups Z(n) and R x Z(n) with the lexicographic order have the extension property and the commutant lifting property. (C) 2001 Academic Press.