THE C*-ALGEBRAS OF THREE-VALUED LOGIC

We show that Grothendieck's functor Ko maps 3-subhomogeneous AF C*-algebras with Hausdorff structure space one-one onto countable Lindenbaum algebras of 3-valued logic. Whereas in the interpretation of Birkhoff and von Neumann propositions arc identified with projections and form an uncountable nondistributive lattice, in our interpretation propositions are unitary equivalence classes of projections, and form a countable MV3 algebra of Chang and Grigolia, alias a 3-valued Lukasiewicz algebra in the sense of Moisil and Monteiro, that is, a Kleene algebra equipped with an operation ∇ obeying x* ∧ ∇x = x* ∧x and x* ∧ ∇x = 1.