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Interpretation of AF \(C^*\)-algebras in Łukasiewicz sentential calculus. (English) Zbl 0597.46059
It is well known that AF \(C^*\)-algebras can be classified completely by the corresponding dimension groups, i.e. the \(K\)-groups with an order unit. Interpreting the \(K\)-group \(K_ 0(A)\) of an AF \(C^*\)-algebra \(A\) as a set of sequences in Łukasiewicz logic, the author gives a criterion for the simplicity of \(A\) in terms of recursion-theoretic properties of \(K_ 0(A)\): If \(A\) is Gödel complete in the sense that the set of consequence of a theory “written in this language” is recursively enumerable but not recursive, then \(A\) cannot be simple. In the case of the CAR algebra the corresponding set of sentences is explicitly worked out.
Reviewer: H.Schröder

MSC:
46L05 General theory of \(C^*\)-algebras
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
03B50 Many-valued logic
03G20 Logical aspects of Łukasiewicz and Post algebras
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)
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[1] Alfsen, E.M, Compact convex sets and boundary integrals, () · Zbl 0209.42601
[2] Bigard, A; Keimel, K; Wolfenstein, S, Groupes et anneaux réticulés, () · Zbl 0384.06022
[3] Bratteli, O, Inductive limits of finite dimensional \(C\^{}\{∗\}\)-algebras, Trans. amer. math. soc., 171, 195-234, (1972) · Zbl 0264.46057
[4] Chang, C.C, Algebraic analysis of many valued logics, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704
[5] Chang, C.C, A new proof of the completeness of the łukasiewicz axioms, Trans. amer. math. soc., 93, 74-80, (1959) · Zbl 0093.01104
[6] Chang, C.C; Keisler, H.J, Model theory, (1977), North-Holland Amsterdam · Zbl 0697.03022
[7] Craig, W, On axiomatizability within a system, J. symbolic logic, 18, 30-32, (1953) · Zbl 0053.20101
[8] Cuntz, J, The internal structure of simple \(C\^{}\{∗\}\)-algebras, (), 85-115
[9] Effros, E.G, Dimensions and \(C\^{}\{∗\}\)-algebras, () · Zbl 0152.33203
[10] Effros, E.G; Handelman, D.E; Shen, C.L, Dimension groups and their affine representation, Amer. J. math., 102, 385-407, (1980) · Zbl 0457.46047
[11] Elliott, G.A, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. algebra, 38, 29-44, (1976) · Zbl 0323.46063
[12] Elliott, G.A, On totally ordered groups, and K0, (), 1-49
[13] Gödel, K, Über formal unentscheidbare Sätze der principia Mathematica und verwandter systeme I, Monatsh. math. phys., 38, 173-198, (1931) · JFM 57.0054.02
[14] Goodearl, K.R, Notes on real and complex \(C\^{}\{∗\}\)-algebras, () · Zbl 1194.16012
[15] Goodearl, K.R; Handelman, D.E, Metric completions of partially ordered abelian groups, Indiana univ. math. J., 29, 861-895, (1980) · Zbl 0455.06012
[16] Grätzer, G, Universal algebra, (1979), Springer-Verlag New York · Zbl 0182.34201
[17] Grigolia, R, Algebraic analysis of łukasiewicz Tarski’s n-valued logical systems, (), 81-92
[18] Haag, R; Kastler, D, An algebraic approach to quantum field theory, J. math. phys., 5, 848-861, (1964) · Zbl 0139.46003
[19] Handelman, D.E, Extensions for AF \(C\^{}\{∗\}\)-algebras and dimension groups, Trans. amer. math. soc., 271, 537-573, (1982) · Zbl 0517.46051
[20] Handelman, D.E; Higgs, D; Lawrence, J, Directed abelian groups, countably continuous rings, and rickart \(C\^{}\{∗\}\)-algebras, J. London math. soc. (2), 21, 193-202, (1980) · Zbl 0449.06013
[21] Kastler, D, Does ergodicity plus locality imply the Gibbs structure?, (), 467-489
[22] Lacava, F, Some properties of ł-algebras and existentially closed ł-algebras, Boll. un. mat. ital. A (5), 16, 360-366, (1979), [Italian] · Zbl 0427.03024
[23] MacLane, S, Categories for the working Mathematician, (1971), Springer-Verlag Berlin
[24] Malinowski, G, Bibliography of łukasiewicz’s logics, (), 189-199
[25] McNaughton, R, A theorem about infinite-valued sentential logic, J. symbolic logic, 16, 1-13, (1951) · Zbl 0043.00901
[26] Monk, J.D, Mathematical logic, (1976), Springer-Verlag New York · Zbl 0354.02002
[27] Mundici, D, Duality between logics and equivalence relations, Trans. amer. math. soc., 270, 111-129, (1982) · Zbl 0497.03018
[28] Mundici, D, Compactness, interpolation and Friedman’s third problem, Ann. of math. logic, 22, 197-211, (1982) · Zbl 0495.03020
[29] Mundici, D, Abstract model theory and nets of \(C\^{}\{∗\}\)-algebras: noncommutative interpolation and preservation properties, (), 351-377
[30] Pedersen, G.K, \(C\^{}\{∗\}\)-algebras and their automorphism groups, (1979), Academic Press London
[31] Rose, A; Rosser, J.B, Fragments of many valued statement calculi, Trans. amer. math. soc., 87, 1-53, (1958) · Zbl 0085.24303
[32] Schwartz, D, Arithmetische theorie der MV-algebren endlicher ordnung, Math. nachr., 77, 65-73, (1977) · Zbl 0409.03038
[33] Tarski, A; Łukasiewicz, J, Investigations into the sentential calculus, (), 38-59
[34] Weinberg, E.C, Free lattice ordered abelian groups, Math. ann., 151, 187-199, (1963) · Zbl 0114.25801
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