Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415239 | Journal of Functional Analysis | 2014 | 9 Pages |
Let A be a commutative unital R-algebra and let Ï be a seminorm on A which satisfies Ï(ab)⩽Ï(a)Ï(b). We apply T. Jacobiʼs representation theorem [10] to determine the closure of a âA2d-module S of A in the topology induced by Ï, for any integer d⩾1. We show that this closure is exactly the set of all elements aâA such that α(a)⩾0 for every Ï-continuous R-algebra homomorphism α:AâR with α(S)â[0,â), and that this result continues to hold when Ï is replaced by any locally multiplicatively convex topology Ï on A. We obtain a representation of any linear functional L:AâR which is continuous with respect to any such Ï or Ï and nonnegative on S as integration with respect to a unique Radon measure on the space of all real-valued R-algebra homomorphisms on A, and we characterize the support of the measure obtained in this way.