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 𝔄 A can be encoded by some theory Φ in the Łukasiewicz calculus, and Φ uniquely determines 𝔄 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.
THE TURING COMPLEXITY OF AF C*-ALGEBRAS WITH LATTICE-ORDERED $K_{0}$ / D. MUNDICI. - STAMPA. - (1987), pp. 256-264. [10.1007/3-540-18170-9_171]
THE TURING COMPLEXITY OF AF C*-ALGEBRAS WITH LATTICE-ORDERED $K_{0}$
MUNDICI, DANIELE
1987
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 𝔄 A can be encoded by some theory Φ in the Łukasiewicz calculus, and Φ uniquely determines 𝔄 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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