dorsal/arxiv
View SchemaNoncommutative Lattices and the Algebra of their Continuous Functions
| Authors | Elisa Ercolessi, Giovanni Landi, Paulo Teotonio-Sobrinho |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9607016 |
| URL | https://arxiv.org/abs/q-alg/9607016 |
| Journal | Rev.Math.Phys. 10 (1998) 439-466 |
Abstract
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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"abstract": "Recently a new kind of approximation to continuum topological spaces has been\nintroduced, the approximating spaces being partially ordered sets (posets) with\na finite or at most a countable number of points. The partial order endows a\nposet with a nontrivial non-Hausdorff topology. Their ability to reproduce\nimportant topological information of the continuum has been the main motivation\nfor their use in quantum physics. Posets are truly noncommutative spaces, or\n{\\it noncommutative lattices}, since they can be realized as structure spaces\nof noncommutative $C^*$-algebras. These noncommutative algebras play the same\nrole of the algebra of continuous functions ${\\cal C}(M)$ on a Hausdorff\ntopological space $M$ and can be thought of as algebras of operator valued\nfunctions on posets. In this article, we will review some mathematical results\nthat establish a duality between finite posets and a certain class of\nC$^*$-algebras. We will see that the algebras in question are all postliminal\napproximately finite dimensional (AF) algebras.",
"arxiv_id": "q-alg/9607016",
"authors": [
"Elisa Ercolessi",
"Giovanni Landi",
"Paulo Teotonio-Sobrinho"
],
"categories": [
"q-alg",
"hep-th",
"math.QA"
],
"journal_ref": "Rev.Math.Phys. 10 (1998) 439-466",
"title": "Noncommutative Lattices and the Algebra of their Continuous Functions",
"url": "https://arxiv.org/abs/q-alg/9607016"
},
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