Abstract
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: 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 $\aleph_3$ (resp., an $\aleph_3$ -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.
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