Simple Bratteli diagrams with a Gödel-incomplete C*-equivalence problem
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In this article, it was shown that the problem of determining whether a pair of abstract simplicial complexes C and A isomorphic to A(C) is not decidable, i.e., it is recursively enumerable, but not necessarily decidable.Abstract:
An abstract simplicial complex is a finite family of subsets of a finite set, closed under subsets. Every abstract simplicial complex C naturally determines a Bratteli diagram and a stable AF-algebra A(C). Consider the following problem: INPUT: a pair of abstract simplicial complexes C and C'; QUESTION: is A(C) isomorphic to A(C')? We show that this problem is Godel 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.read more
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References
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Interpretation of AF C∗-algebras in Łukasiewicz sentential calculus
TL;DR: In this paper, a non-simplicite critere pour la nonsimplicity d'une AF Cα-algebre #7B-U en termes de proprietes de la theorie de la recursivite du groupe dimension K 0 (#7B -U) was presented.
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Inductive limits of finite dimensional C*-algebras
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Combinatorial Convexity and Algebraic Geometry
TL;DR: In this article, the boundary complex of polytopes and polyhedral sets is discussed. But the authors do not consider the relation between the two types of sets and do not provide a classification of the two sets.