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    Positive representations of C0(X)C_0(X). I

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    We introduce the notion of a positive spectral measure on a Οƒ\sigma-algebra, taking values in the positive projections on a Banach lattice. Such a measure generates a bounded positive representation of the bounded measurable functions. If XX is a locally compact Hausdorff space, and Ο€\pi is a positive representation of C0(X)C_0(X) on a KB-space, then Ο€\pi is the restriction to C0(X)C_0(X) of such a representation generated by a unique regular positive spectral measure on the Borel Οƒ\sigma-algebra of XX. The relation between a positive representation of C0(X)C_0(X) on a Banach lattice and -- if it exists -- a generating positive spectral measure on the Borel Οƒ\sigma-algebra is further investigated; here and elsewhere phenomena occur that are specific for the ordered context.Comment: There is now a direct proof of the existence of a generating regular positive spectral measure in the case of KB-spaces, without resorting to the Banach space theory. References to the existing literature on the Banach space case have been added, and perspectives for future research are now given. 24 pages, to appear in Ann. Funct. Ana
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