The Category of Bratteli Diagrams

Amini, Massoud ; Elliott, George A. ; Golestani, Nasser

Canadian journal of mathematics, Tome 67 (2015) no. 5, pp. 990-1023

Résumé

A category structure for Bratteli diagrams is proposed and a functor from the category of $\text{AF}$ algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli’s notion of equivalence, we obtain in particular a functorial formulation of Bratteli’s classification of $\text{AF}$ algebras (and at the same time, of Glimm’s classification of $\text{UHF}$ algebras). It is shown that the three approaches to classification of $\text{AF}$ algebras, namely, through Bratteli diagrams, $\text{K}$ -theory, and a certain natural abstract classifying category, are essentially the same from a categorical point of view.

Détail

DOI : 10.4153/CJM-2015-001-8

Mots-clés : 46L05, 46L35, 46M15, C*-algebra, category, functor, AF algebra, dimension group, Bratteli diagram

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