Gödel incompleteness in AF C*-algebras
Daniele Mundici, Constantine Tsinakis · Forum Mathematicum · 2008
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 a 1 + ⋯ + a k ≤ b 1 + ⋯ + b l satisfied by G ( A ), for any a i , b j ∈ D . By Elliott's classification, 𝒫 together with the list of all sums a 1 + ⋯ + a k ∈ 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 .