## 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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### References:

 [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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