Lifting defects for nonstable K_0-theory of exchange rings and C*-algebras

TL;DR

This paper investigates the limitations of lifting nonstable K_0-theory functors from monoids to exchange rings and C*-algebras, revealing fundamental obstructions and constructing specific counterexamples.

Contribution

It proves the non-existence of certain functors that lift K_0-theory from monoids to rings and C*-algebras, and constructs explicit counterexamples demonstrating these limitations.

Findings

No functor from simplicial monoids to exchange rings can be equivalent to the identity.

No functor from simplicial monoids to C*-algebras of real rank 0 can be equivalent to the identity.

Existence of a unital exchange ring with specific properties whose V-monoid is not isomorphic to that of any C*-algebra of real rank 0 or regular ring.

Abstract

The assignment (nonstable K_0-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: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp., von Neumann regular rings), such that VF is equivalent to the identity. (3) There is a {0,1}^3-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 from an earlier paper (larders, lifters, CLL), we deduce that there exists a unital exchange ring of cardinality aleph three (resp., an aleph three-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 and V(R) is not isomorphic to V(B) for any ring B which is either a C*-algebra of real rank 0 or a regular ring.