Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Noncommutative Lattices and the Algebra of their Continuous Functions   Recently a new kind of approximation to continuum topological spaces has been
introduced, the approximating spaces being partially ordered sets (posets) with
a finite or at most a countable number of points. The partial order endows a
poset with a nontrivial non-Hausdorff topology. Their ability to reproduce
important topological information of the continuum has been the main motivation
for their use in quantum physics. Posets are truly noncommutative spaces, or
{\it noncommutative lattices}, since they can be realized as structure spaces
of noncommutative $C^*$-algebras. These noncommutative algebras play the same
role of the algebra of continuous functions ${\cal C}(M)$ on a Hausdorff
topological space $M$ and can be thought of as algebras of operator valued
functions on posets. In this article, we will review some mathematical results
that establish a duality between finite posets and a certain class of
C$^*$-algebras. We will see that the algebras in question are all postliminal
approximately finite dimensional (AF) algebras.

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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: funct-an
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: plasm-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: astro-ph