Gelfand-Neumark duality states that the category of compact Hausdorff spaces and continuous functions is dually equivalent to the category of commutative C*-algebra and ℓ-algebra homomorphisms. The real setting of this duality was established by Stone. Banaschewski called the real analogue of C*-algebras Stone rings. Many mathematicians studied larger categories of Stone rings, for example bal, the category of bounded Archimedean ℓ-algebras.
In this paper, we study the category baf of bounded Archimedean f-rings. We illustrate the categorical connection between baf and bal. Using an ℓ-group theoretic tensor product, that we extend to ℓ-rings, we describe the reflection of an object of baf in bal. We also study some interesting subcategories of baf originally described by Madden and Schwartz for partially ordered rings. We show that the subclass of ℓ-clean rings in baf corresponds dually to Stone spaces. Finally we study C( X,A) when A is in baf and investigate the properties inherited by C(X,A) from A and vice versa.
|School:||New Mexico State University|
|School Location:||United States -- New Mexico|
|Source:||DAI-B 77/11(E), Dissertation Abstracts International|
|Keywords:||Bounded Archimedean f-rings, Bounded Inversion, F-Tensor Product, Gelfand Duality, Lattice Ordered Rings, Ordered Tensor Products|
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