Gödel incompleteness in AF C*-algebras
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Daniele Mundici
Abstract
For any (possibly, non-unital) AF C*-algebra A with comparability of projections, let D(A) be the Elliott partial monoid of A, and G(A) the dimension group of A with scale D(A). For D ⊆ D(A) a generating set of G(A) let 𝒫 be the set of all formal inequalities a1 + ⋯ + ak ≤ b1 + ⋯ + bl satisfied by G(A), for any ai, bj ∈ D. By Elliott's classification, 𝒫 together with the list of all sums a1 + ⋯ + ak ∈ D(A) uniquely determines A. Can 𝒫 be Gödel incomplete, i.e., effectively enumerable but undecidable? We give a negative answer in case D is finite, and a positive answer in the infinite case. We also show that the range of the map A ↦ D(A) precisely consists of all countable partial abelian monoids satisfying the following three conditions: (i) a + b = a + c ⇒ b = c, (ii) a + b = 0 ⇒ a = b = 0 and (iii) ∀a, b ∈ E ∃c ∈ E such that either a + c = b or b + c = a.
© de Gruyter 2008
Articles in the same Issue
- Inequalities for Euler's gamma function
- Integers without divisors from a fixed arithmetic progression
- Sharp results on the integrability of the derivative of an interpolating Blaschke product
- The Gauss map of pseudo-algebraic minimal surfaces
- Gödel incompleteness in AF C*-algebras
- Quasi-regular Dirichlet forms on Riemannian path and loop spaces
- Divided differences and generalized Taylor series
- A regularity result for a class of degenerate Yang-Mills connections in critical dimensions
Articles in the same Issue
- Inequalities for Euler's gamma function
- Integers without divisors from a fixed arithmetic progression
- Sharp results on the integrability of the derivative of an interpolating Blaschke product
- The Gauss map of pseudo-algebraic minimal surfaces
- Gödel incompleteness in AF C*-algebras
- Quasi-regular Dirichlet forms on Riemannian path and loop spaces
- Divided differences and generalized Taylor series
- A regularity result for a class of degenerate Yang-Mills connections in critical dimensions
