Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras

The assignment (nonstable K0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (i) There is no functor Γ, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that V ◦Γ ∼= id. (ii) There is no functor Γ, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that V ◦Γ ∼= id. (iii) There is a {0, 1}-indexed commutative diagram ~ D of simplicial monoids that can be lifted, with respect to the functor V, by exchange rings and by C*-algebras of real rank 1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank 0. By using categorical tools (larders, lifters, CLL) from a recent book from the author with P. Gillibert, we deduce that there exists a unital exchange ring of cardinality א3 (resp., an א3-separable unital C*-algebra of real rank 1) R, with stable rank 1 and index of nilpotence 2, such that V(R) is the positive cone of a dimension group but it is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.