Riesz* homomorphisms on pre-Riesz spaces consisting of continuous functions

作者：van Imhoff Hendrik^*

来源：Positivity, 2018, 22(2): 425-447.

DOI：10.1007/s11117-017-0519-4

• Partially ordered vector space
• Order dense subspace
• Pre-Riesz space
• Riesz* homomorphism
• Weighted composition operator
• Automorphism group
• Smooth manifold
• Sobolev space

In the theory of operators on a Riesz space (vector lattice), an important result states that the Riesz homomorphisms (lattice homomorphisms) on C(X) are exactly the weighted composition operators. We extend this result to Riesz* homomorphisms on order dense subspaces of C(X). On those subspace we consider and compare various classes of operators that extend the notion of a Riesz homomorphism. Furthermore, using the weighted composition structure of Riesz* homomorphisms we obtain several results concerning bijective Riesz* homomorphisms. In particular, we characterize the automorphism group for order dense subspaces of C(X). Lastly, we develop a similar theory for Riesz* homomorphisms on subspace of , for a locally compact Hausdorff space X, and apply it to smooth manifolds and Sobolev spaces.

• 出版日期2018-4