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

✍ Scribed by Ramón Bruzual; Marisela Domı́nguez

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
158 KB
Volume
185
Category
Article
ISSN
0022-1236

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✦ Synopsis

Let 0 be a locally compact abelian ordered group. We say that 0 has the extension property if every operator valued continuous positive definite function on an interval of 0 has a positive definite extension to the whole group and we say that 0 has the commutant lifting property if a natural extension of the commutant lifting theorem holds on 0. 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 1 has the extension property and satisfies an archimedean condition then 0=1_Z with the lexicographic order has the extension property. As an application we obtain that the groups Z n and R_Z n with the lexicographic order have the extension property and the commutant lifting property.

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