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Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem

Author: Daniele Mundici
Journal: Trans. Amer. Math. Soc. 356 (2004), 1937-1955
MSC (2000): Primary 46L35, 06F20, 20F10, 03D40, 52B20
Published electronically: June 24, 2003
MathSciNet review: 2031047
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Abstract: An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex $\mathcal{C}$ naturally determines a Bratteli diagram and a stable AF-algebra $A(\mathcal{C})$. Consider the following problem:

INPUT: a pair of abstract simplicial complexes $\mathcal{C}$ and $\mathcal{C}'$;

QUESTION: is $A(\mathcal{C})$ isomorphic to $A(\mathcal{C}')$?

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.

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Additional Information

Daniele Mundici
Affiliation: Department of Mathematics “Ulisse Dini”, University of Florence, Viale Morgagni 67/A, 50134 Florence, Italy

Keywords: Isomorphism of Bratteli diagrams, stable AF-algebra, Elliott's classification, Markov undecidability theorem, $C^{*}$-equivalence, De Concini-Procesi theorem for toric varieties
Received by editor(s): March 12, 2002
Received by editor(s) in revised form: March 31, 2003
Published electronically: June 24, 2003
Additional Notes: Partially supported by MURST Project on Logic
Article copyright: © Copyright 2003 American Mathematical Society

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