Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. W. M. Beynon, Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canad. J. Math., 29:243-254, 1977. MR 55:10350
  • 2. A. Bigard, K. Keimel, and S. Wolfenstein, Groupes et Anneaux Réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, New York, 1977. MR 58:27688
  • 3. O. Bratteli, Inductive limits of finite dimensional $C^*$-algebras, Trans. Amer. Math. Soc., 171:195-234, 1972. MR 47:844
  • 4. O. Bratteli, P. Jorgensen, K. H. Kim, and F. Roush, Non-stationarity of isomorphism between AF-algebras defined by stationary Bratteli diagrams, Ergodic Theory and Dynamical Systems, 20:1639-1656, 2000. MR 2001k:46104
  • 5. O. Bratteli, P. Jorgensen, K. H. Kim, and F. Roush, Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups, Ergodic Theory and Dynamical Systems, 21: 1625-1655, 2001. MR 2002h:46088
  • 6. E. G. Effros, Dimensions and $C^*$-algebras, C.B.M.S. Regional Conference Series in Math., Vol. 46, Amer. Math. Soc., Providence, RI, 1981. MR 84k:46042
  • 7. G. A. Elliott and D. Mundici, A characterisation of lattice-ordered abelian groups, Math. Zeitschrift 213:179-185, 1993. MR 94e:06010
  • 8. G. Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, Vol. 168, Springer-Verlag, New York, 1996. MR 97i:52012
  • 9. A. M. W. Glass and J. J. Madden, The word problem versus the isomorphism problem, J. London Math. Soc., (2), 30:53-61, 1984. MR 86i:03059
  • 10. A. J. Goldman and A. W. Tucker, Polyhedral convex cones. In Linear equalities and related systems, pp. 19-40, Annals of Mathematics Studies, Vol. 38, Princeton University Press, Princeton, NJ, 1956, MR 19:446g
  • 11. K. Goodearl, Notes on Real and Complex $C^{*}$-algebras, Birkhäuser, Boston, 1982 (Volume 5 of Shiva Mathematics Series). MR 85d:46079
  • 12. V. Marra, Every abelian $\ell$-group is ultrasimplicial, Journal of Algebra, 225:872-884, 2000. MR 2001c:06020
  • 13. D. Mundici, Interpretation of AF $C^{*}$-algebras in \Lukasiewicz sentential calculus, J. Functional Analysis, 65:15-63, 1986. MR 87k:46146
  • 14. D. Mundici, Satisfiability in many-valued sentential logic is NP-complete, Theoretical Computer Science, 52:145-153, 1987. MR 89a:68076
  • 15. D. Mundici, Farey stellar subdivisions, ultrasimplicial groups, and $K_0$ of AF $C^*$-algebras, Advances in Math., 68:23-39, 1988. MR 89d:46072
  • 16. D. Mundici, Classes of ultrasimplicial lattice-ordered abelian groups, Journal of Algebra, 213:596-603, 1999. MR 2000d:06023
  • 17. D. Mundici and G. Panti, The equivalence problem for Bratteli diagrams, Technical Report no. 259, University of Siena (Italy), 1993. Unpublished, 7 pp.
  • 18. T. Oda, Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin, 1988. MR 88m:14038

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L35, 06F20, 20F10, 03D40, 52B20

Retrieve articles in all journals with MSC (2000): 46L35, 06F20, 20F10, 03D40, 52B20

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

American Mathematical Society