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Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem |
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| Authors: | Daniele Mundici |
| Affiliation: | Department of Mathematics ``Ulisse Dini', University of Florence, Viale Morgagni 67/A, 50134 Florence, Italy |
| Abstract: | An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex  naturally determines a Bratteli diagram and a stable AF-algebra  . Consider the following problem: INPUT: a pair of abstract simplicial complexes QUESTION: is We show that this problem is Gödel incomplete, i.e., it is recursively enumerable but not decidable. This result is in sharp contrast with the recent decidability result by Bratteli, Jorgensen, Kim and Roush, for the isomorphism problem of stable AF-algebras arising from the iteration of the same positive integer matrix. For the proof we use a combinatorial variant of the De Concini-Procesi theorem for toric varieties, together with the Baker-Beynon duality theory for lattice-ordered abelian groups, Markov's undecidability result, and Elliott's classification theory for AF-algebras. |
| Keywords: | Isomorphism of Bratteli diagrams stable AF-algebra Elliott's classification Markov undecidability theorem $C^{*}$-equivalence De Concini-Procesi theorem for toric varieties |
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