Archimedean ℓ-groups with strong unit: cozero-sets and coincidence of types of ideals
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Papiya Bhattacharjee
Abstract
W* is the category of the title. For G ∈ W*, we have the canonical compact space YG, and Yosida representation G ≤ C(YG), thus, for g ∈ G, one has the cozero-set coz(g) in YG. The ideals at issue in G include the principal ideals and polars, G(g) and g⊥⊥, respectively, and the W*-kernels of W*-morphisms from G. The “coincidences of types” include these properties of G: (M) Each G(g) = g⊥⊥; (Y) Each G(g) is a W*-kernel; (CR) Each g⊥⊥ is a W*-kernel (iff each coz(g) is regular open). For each of these, we give numerous “rephrasings” and examples, and note that (M) = (Y) ∩ (CR). This paper is a companion to a paper in preparation by the present authors, which includes the present thrust in contexts less restrictive and more algebraic. Here, the focus on W* brings topology to bear, and sharpens the view.
B. Wynne was supported by PSC-CUNY Research Award #66070-00 54
Acknowledgement
We thank the anonymous referee for their comments, which helped us to improve the exposition of the paper.
Communicated by Roberto Giuntini
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