K-Theory for Certain C ∗ -Algebras

Cuntz, Joachim

doi:10.2307/1971137

Cited by 369 publications

(327 citation statements)

References 8 publications

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“…0 An interesting situation in which separativity occurs is the case of an a-simple 'purely infinite' monoid, as follows. This is a monoid version of an argument of Cuntz [20 Although separativity for a ring R is an 'external' condition in that it involves all the modules from F P(R), it is equivalent to a corresponding 'internal' version involving direct summands of R in case R is an exchange ring (Corollary 2.9). En route to proving this, we give the main reduction step as a lemma that will be used again later.…”

Section: Proposition 23 Any Directly Finite Separative Ring R Is Stmentioning

confidence: 98%

Separative cancellation for projective modules over exchange rings

et al. 1998

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-A separative ring is one whose finitely generated projective modules satisfy the property A EB A � A EB B � B EB B ==} A ,...., B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separate exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and Rf I both separative, then R is separative. SEPARATIVE CANCELLATION FOR PROJECTIVE MODULES OVER EXCHANGE RINGSP. ARA, K.R. GOODEARL, K.C. O'MEARA AND E. PARDO ABSTRACT. A separative ring is one whose finitely generated projective modules satisfy the property A EB A ~ A EB B ~ B EB B ~ A ~ B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under extensions of ideals by factor rings. That is, if an exchange ring R has an ideal I with I and R/ I both separative, then R is separative.

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Section: Proposition 23 Any Directly Finite Separative Ring R Is Stmentioning

confidence: 98%

Separative cancellation for projective modules over exchange rings

et al. 1998

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“…(a) Many purely infinite simple C*-algebras are known; for instance, see [1], [11], [12], [16], [17], [18], [21], [25], [26], [37]. (b) If V is an infinite dimensional vector space over k, then End k (V ) modulo its unique maximal ideal M is purely infinite.…”

Section: Definitions 12mentioning

confidence: 99%

K0 of Purely Infinite Simple Regular Rings

Ara¹,

Goodearl²,

Pardo³

2002

K-Theory

130

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Abstract. We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if R is a purely infinite simple ring, then K 0 (R) + = K 0 (R), the monoid of isomorphism classes of finitely generated projective R-modules is isomorphic to the monoid obtained from K 0 (R) by adjoining a new zero element, and K 1 (R) is the abelianization of the group of units of R. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we compute the Grothendieck groups of these examples. In particular, we prove that every countable abelian group is isomorphic to K 0 of some purely infinite simple regular ring. Finally, some known examples are analyzed within this framework.

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“…Suppose that A is simple and infinite. Then, by [7,Proposition 1.5], A contains two isometries with orthogonal ranges and so, by [16,Proposition 6.5], sr(A) = ∞. In the case of finite, simple C*-algebras the following is known: Whenever A is simple and stably finite and B is a UHF-algebra, the tensor product A⊗B has stable rank one; see [19].…”

Section: Introductionmentioning

confidence: 99%

On the stable rank of simple C*-algebras

Villadsen

1999

J. Amer. Math. Soc.

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K-Theory for Certain C ∗ -Algebras

Cited by 369 publications

References 8 publications

Separative cancellation for projective modules over exchange rings

Separative cancellation for projective modules over exchange rings

K0 of Purely Infinite Simple Regular Rings

On the stable rank of simple C*-algebras

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