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Journal of Operator Theory

Volume 76, Issue 1, Summer 2016  pp. 33-56.

Regular representations of lattice ordered semigroups

Authors:  Boyu Li
Author institution: Pure Mathematics Department, University of Waterloo, Waterloo, ON, Canada N2L--3G1

Summary:  We establish a necessary and sufficient condition for a representation of a lattice ordered semigroup to be regular, in the sense that certain extensions are completely positive definite. This result generalizes a theorem due to Brehmer where the lattice ordered group was taken to be $\mathbb{Z}_+^\Omega$. As an immediate consequence, we prove that contractive Nica-covariant representations are regular. We also introduce an analog of commuting row contractions on a lattice ordered group and show that such a representation is regular.

Keywords:  Nica-covariant, regular dilation, positive definite, lattice ordered group

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