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

Mundici, Daniele

doi:10.1090/S0002-9947-03-03353-1

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

Trans. Amer. Math. Soc. 356 (2004), 1937-1955

DOI: https://doi.org/10.1090/S0002-9947-03-03353-1

Published electronically: June 24, 2003

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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.

References

W. M. Beynon, Applications of duality in the theory of finitely generated lattice-ordered abelian groups, Canadian J. Math. 29 (1977), no. 2, 243–254. MR 437420, DOI 10.4153/CJM-1977-026-4
Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR 0552653
Ola Bratteli, Inductive limits of finite dimensional $C^{\ast }$-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234. MR 312282, DOI 10.1090/S0002-9947-1972-0312282-2
Ola Bratteli, Palle E. T. Jørgensen, Ki Hang Kim, and Fred Roush, Non-stationarity of isomorphism between AF algebras defined by stationary Bratteli diagrams, Ergodic Theory Dynam. Systems 20 (2000), no. 6, 1639–1656. MR 1804950, DOI 10.1017/S0143385700000912
Ola Bratteli, Palle E. T. Jorgensen, Ki Hang Kim, and Fred Roush, Decidability of the isomorphism problem for stationary AF-algebras and the associated ordered simple dimension groups, Ergodic Theory Dynam. Systems 21 (2001), no. 6, 1625–1655. MR 1869063, DOI 10.1017/S014338570100178X
Edward G. Effros, Dimensions and $C^{\ast }$-algebras, CBMS Regional Conference Series in Mathematics, vol. 46, Conference Board of the Mathematical Sciences, Washington, D.C., 1981. MR 623762
George A. Elliott and Daniele Mundici, A characterisation of lattice-ordered abelian groups, Math. Z. 213 (1993), no. 2, 179–185. MR 1221712, DOI 10.1007/BF03025717
Günter Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Mathematics, vol. 168, Springer-Verlag, New York, 1996. MR 1418400, DOI 10.1007/978-1-4612-4044-0
A. M. W. Glass and James J. Madden, The word problem versus the isomorphism problem, J. London Math. Soc. (2) 30 (1984), no. 1, 53–61. MR 760872, DOI 10.1112/jlms/s2-30.1.53
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
K. R. Goodearl, Notes on real and complex $C^{\ast }$-algebras, Shiva Mathematics Series, vol. 5, Shiva Publishing Ltd., Nantwich, 1982. MR 677280
Vincenzo Marra, Every abelian $l$-group is ultrasimplicial, J. Algebra 225 (2000), no. 2, 872–884. MR 1741567, DOI 10.1006/jabr.1999.8163
Daniele Mundici, Interpretation of AF $C^\ast$-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), no. 1, 15–63. MR 819173, DOI 10.1016/0022-1236(86)90015-7
Daniele Mundici, Satisfiability in many-valued sentential logic is NP-complete, Theoret. Comput. Sci. 52 (1987), no. 1-2, 145–153. MR 918116, DOI 10.1016/0304-3975(87)90083-1
Daniele Mundici, Farey stellar subdivisions, ultrasimplicial groups, and $K_0$ of AF $C^*$-algebras, Adv. in Math. 68 (1988), no. 1, 23–39. MR 931170, DOI 10.1016/0001-8708(88)90006-0
Daniele Mundici, Classes of ultrasimplicial lattice-ordered abelian groups, J. Algebra 213 (1999), no. 2, 596–603. MR 1673471, DOI 10.1006/jabr.1998.7679
D. Mundici and G. Panti, The equivalence problem for Bratteli diagrams, Technical Report no. 259, University of Siena (Italy), 1993. Unpublished, 7 pp.
Tadao Oda, Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15, Springer-Verlag, Berlin, 1988. An introduction to the theory of toric varieties; Translated from the Japanese. MR 922894