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) |
Keywords:
AF \(C^*\)-algebras; dimension groups; K-groups with an order; Łukasiewicz logic; recursion-theoretic properties of \(K_ 0(A)\); Gödel complete; CAR algebraReferences:
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