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Bounded Archimedean l-algebras and Gelfand-Neumark-Stone duality

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Guram Bezhanishvili, Patrick J. Morandi, Bruce Olberding

By Gelfand-Neumark duality, the category $C^*Alg$
of commutative $C^*$-algebras is dually equivalent to the category of
compact Hausdorff spaces, which by Stone duality, is also dually
equivalent to the category $ubal$ of uniformly complete bounded
Archimedean $\ell$-algebras. Consequently, $C^*Alg$ is equivalent
to $ubal$, and this equivalence can be described through
complexification.

In this article we study $ubal$ within the larger category $bal$ of
bounded Archimedean $\ell$-algebras. We show that $ubal$ is the
smallest nontrivial reflective subcategory of $bal$, and that $ubal$
consists of exactly those objects in $bal$ that are epicomplete, a fact
that includes a categorical formulation of the Stone-Weierstrass theorem
for $bal$. It follows that $ubal$ is the unique nontrivial reflective
epicomplete subcategory of $bal$. We also show that each nontrivial
reflective subcategory of $bal$ is both monoreflective and
epireflective, and exhibit two other interesting reflective
subcategories of $bal$ involving Gelfand rings and square closed rings.

Dually, we show that Specker ${\mathbb R}$-algebras are precisely the
co-epicomplete objects in $bal$. We prove that the category $spec$ of
Specker $\mathbb R$-algebras is a mono-coreflective subcategory of
$bal$ that is co-epireflective in a mono-coreflective subcategory of
$bal$ consisting of what we term $\ell$-clean rings, a version of clean
rings adapted to the order-theoretic setting of $bal$.

We conclude the article by discussing the import of our results in the
setting of complex $*$-algebras through complexification.

Keywords:
Ring of continuous
real-valued functions, l-ring, l-algebra, uniform
completeness, Stone-Weierstrass theorem, commutative $C^*$-algebra,
compact Hausdorff space, Gelfand-Neumark-Stone duality

2010 MSC:
06F25; 13J25; 54C30

*Theory and Applications of Categories,*
Vol. 28, 2013,
No. 16, pp 435-475.

Published 2013-07-15.

http://www.tac.mta.ca/tac/volumes/28/16/28-16.dvi

http://www.tac.mta.ca/tac/volumes/28/16/28-16.ps

http://www.tac.mta.ca/tac/volumes/28/16/28-16.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/16/28-16.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/28/16/28-16.ps

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