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

Abstract

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 K0(A) uniquely determines A up to isomorphism. AF-algebras with lattice-ordered K0, (AFℓ-algebras) have a preeminent role in the AF-algebraic literature. The Elliott classifier E(A) of any AFℓ-algebra A—i.e., the unit interval of K0(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 AFℓ-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 ϕA 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 ϕA⪯ψA?IsϕA={0}? Is ϕA central ? For any AFℓ-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 AFℓ-algebras in the literature, including the Behncke-Leptin algebras Am,n, the Farey AFℓ-algebra M1 and all its principal quotients, the CAR algebra and Glimm's universal UHF algebra, and every Effros-Shen algebra Fθ for θ∈[0,1]∖Q a real algebraic integer, or for θ=1/e. For any fixed n=1,2,… we consider the class Kn of AFℓ-algebras presented by generators X1,…,Xn and relations θ1=0,…,θm=0, for an arbitrary finite set Θ={θ1,…,θm} of Ł∞-formulas in the variables X1,…,Xn. We prove that all decision problems studied in this paper are polytime decidable for Kn—uniformly over the presentation Θ.