DEDEKIND COMPLETIONS OF BOUNDED ARCHIMEDEAN l-ALGEBRAS

作者：Bezhanishvili Guram^*; Morandi Patrick J; Olberding Bruce

来源：Journal of Algebra and Its Applications, 2013, 12(1): 1250139.

DOI：10.1142/S0219498812501393

• l-ring
• l-algebra
• Baer ring
• uniform completion
• Dedekind completion
• essential closure
• injective hull
• C*-algebra
• compact Hausdorff space
• Gelfand-Neumark-Stone duality
• extremally disconnected space
• projective cover

All algebras considered in this paper are commutative with 1. Let bal be the category of bounded Archimedean l-algebras. We investigate Dedekind completions and Dedekind complete algebras in bal. We give several characterizations for A epsilon bal to be Dedekind complete. Also, given A, B epsilon bal, we give several characterizations for B to be the Dedekind completion of A. We prove that unlike general Gelfand-Neumark-Stone duality, the duality for Dedekind complete algebras does not require any form of the Stone-Weierstrass Theorem. We show that taking the Dedekind completion is not functorial, but that it is functorial if we restrict our attention to those A bal that are Baer rings. As a consequence of our results, we give a new characterization of when A epsilon bal is a C*-algebra. We also show that A is a C*-algebra if and only if A is the inverse limit of an inverse family of clean C*-algebras. We conclude the paper by discussing how to derive Gleason's theorem about projective compact Hausdorff spaces and projective covers of compact Hausdorff spaces from our results.

• 出版日期2013-2