Abstract
Approximately finite-dimensional (AF) C⋆-algebras were introduced in 1972 by Bratteli, generalizing earlier work of Glimm and Dixmier. In a recent paper, the author presents a natural one-one correspondence between Lindenbaum algebras of the infinite-valued sentential calculus of Łukasiewicz, and AF C⋆-algebras whose Grothendieck group (KO) is lattice-ordered. Thus, any such algebra \(\mathfrak{A}\) can be encoded by some theory Φ in the Łukasiewicz calculus, and Φ uniquely determines \(\mathfrak{A}\), up to isomorphism. In the present paper, Glimm's universal UHF algebra, the Canonical Anticommutation Relation (CAR) algebra, and the Effros-Shen algebras corresponding to quadratic irrationals are explicitly coded by theories whose decision problems are solvable in deterministic polynomial time.
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