Extensions of Operator Valued Positive Definite Functions and Commutant Lifting on Ordered Groups

Let Ω be a locally compact abelian ordered group. We say that Ω has the extension property if every operator valued continuous positive definite function on an interval of Ω has a positive definite extension to the whole group and we say that Ω has the commutant lifting property if a natural extension of the commutant lifting theorem holds on Ω. 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 isometries. It is proved that if a group has the extension property and satisfies an archimedean condition then it has the commutant lifting property. It is also proved that if the ordered group Γ has the extension property and satisfies an archimedean condition then Ω=Γ×Z with the lexicographic order has the extension property. As an application we obtain that the groups Zn and R×Zn with the lexicographic order have the extension property and the commutant lifting property.