Simple Bratteli Diagrams with a Gödel-incomplete C*-equivalence Problem

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 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.