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

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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:

  1. (i)

    There is no functor Γ, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that V ∘ Γ ≅ id.

  2. (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.

  3. (iii)

    There is a {0,1}3-indexed commutative diagram \({\vec{D}}\) 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 \(\aleph_3\) (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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Authors and Affiliations

  1. LMNO, CNRS UMR 6139, Département de Mathématiques, Université de Caen, 14032, Caen Cedex, France

    Friedrich Wehrung

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Wehrung, F. Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras. Algebr Represent Theor 16, 553–589 (2013). https://doi.org/10.1007/s10468-011-9319-x

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