AF-algebras with lattice-ordered K0: Logic and computation

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Following Bratteli's original definition, an AF-algebra A is the norm closure of the union of an ascending sequence of finite-dimensional C ⁎ -algebras, all with the same unit. Elliott proved that the unital dimension (Grothendieck) group K 0 ( A ) uniquely determines A up to isomorphism. AF-algebras with lattice-ordered K 0 , ( AFℓ-algebras ) have a preeminent role in the AF-algebraic literature. The Elliott classifier E ( A ) of any AF ℓ -algebra A —i.e., the unit interval of K 0 ( A ) , or equivalently, the countable set of Murray-von Neumann equivalence classes of projections of A —has the structure of a countable MV-algebra. Every countable MV-algebra arises as E ( A ) for some AF ℓ -algebra A . Since E ( A ) is the Lindenbaum algebra of some countable set of formulas in Łukasiewicz logic Ł ∞ , every Ł ∞ -formula ϕ naturally codes an equivalence class ϕ A of projections in A . The deductive-algorithmic machinery of Ł ∞ can be thus applied to decide, among many others, the following basic problems, with ⪯ the Murray-von Neumann order: Is ϕ A ⪯ ψ A ? Is ϕ A = { 0 } ? Is ϕ A central ? For any AF ℓ -algebra A we construct mutual polytime reductions of the first two problems. The third problem is polytime reducible to both problems. If E ( A ) is finitely generated and A either has no quotient isomorphic to C or is simple, then the first two problems are polytime reducible to the third one. The complexity of these problems is polytime for many AF ℓ -algebras in the literature, including the Behncke-Leptin algebras A m , n , the Farey AF ℓ -algebra M 1 and all its principal quotients, the CAR algebra and Glimm's universal UHF algebra, and every Effros-Shen algebra F θ for θ ∈ [ 0 , 1 ] ∖ Q a real algebraic integer, or for θ = 1 / e . For any fixed n = 1 , 2 , … we consider the class K n of AF ℓ -algebras presented by generators X 1 , … , X n and relations θ 1 = 0 , … , θ m = 0 , for an arbitrary finite set Θ = { θ 1 , … , θ m } of Ł ∞ -formulas in the variables X 1 , … , X n . We prove that all decision problems studied in this paper are polytime decidable for K n —uniformly over the presentation Θ.