Lifting Defects for Nonstable K0-theory of Exchange Rings and C*-algebras
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- 作者：Friedrich Wehrung (1)
- 关键词：Ring ; Exchange property ; Regular ; C* ; algebra ; Real rank ; Stable rank ; Index of nilpotence ; Semiprimitive ; V ; semiprimitive ; Weakly V ; semiprimitive ; Simplicial monoid ; Dimension group ; Commutative monoid ; Order ; unit ; O ; ideal ; Refinement property ; Nonstable ; K ; theory ; Idempotent ; Orthogonal ; Projection ; Functor ; Diagram ; Lifting ; Premeasure ; Measure ; Larder ; Lifter ; Condensate ; CLL ; 19A49 ; 46L80 ; 16B50 ; 16B70 ; 16E20 ; 16E50 ; 16N60 ; 18A30 ; 18C35 ; 06B20 ; 08B20 ; 03E05
- 刊名：Algebras and Representation Theory
- 出版年：2013
- 出版时间：April 2013
- 年：2013
- 卷：16
- 期：2
- 页码：553-589
- 全文大小：779KB
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1. LMNO, CNRS UMR 6139, D茅partement de Math茅matiques, Universit茅 de Caen, 14032, Caen Cedex, France - ISSN：1572-9079
文摘
The assignment (nonstable K0-theory), that to a ring聽R associates the monoid聽V(鈥?em class="a-plus-plus">R鈥? of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over聽R, extends naturally to a functor. We prove the following lifting properties of that functor: There is no functor 螕, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that V 鈭樷€壩撯€夆墔 id. There is no functor 螕, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank聽0 (resp., von Neumann regular rings), such that V 鈭樷€壩撯€夆墔 id. There is a {0,1}3-indexed commutative diagram聽 ${\vec{D}}$ of simplicial monoids that can be lifted, with respect to the functor聽V, by exchange rings and by C*-algebras of real rank聽1, but not by semiprimitive exchange rings, thus neither by regular rings nor by C*-algebras of real rank聽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聽 $\aleph_3$ (resp., an $\aleph_3$ -separable unital C*-algebra of real rank聽1)聽R, with stable rank聽1 and index of nilpotence聽2, such that聽V(鈥?em class="a-plus-plus">R鈥? is the positive cone of a dimension group but it is not isomorphic to聽V(鈥?em class="a-plus-plus">B鈥? for any ring聽B which is either a C*-algebra of real rank聽0 or a regular ring.