On the stable rank of simple C*-algebras

Villadsen, Jesper

doi:10.1090/s0894-0347-99-00314-8

Cited by 91 publications

(82 citation statements)

References 18 publications

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“…For a unital simple C -algebra A, Theorem 4.2 indicates that the only case when Dur A might not be 1 is when A is stably finite and has stable rank greater than 1. The only example of this that we know so far is given by Villadsen [1999].…”

Section: David Hartleymentioning

confidence: 99%

“…Fix an integer n > 1. Let A D lim k!1 .A k ; k / be the unital simple AH-algebra with tsr A D n constructed by Villadsen [1999]. Then A 1 D C.D n /.…”

Section: David Hartleymentioning

confidence: 99%

“…k;j /˝p k;j for all f 2 A k , where p k;1 is a trivial rank-1 projection, A kC1 D k .id A k /M .r.k/ .C.X k // k .id A k / (for some large r.n/) for some spaces X k , and k;j W X kC1 ! X k is a continuous map (these are 1 i C1 and some point evaluations as denoted in [Villadsen 1999[Villadsen , p. 1092). Clearly A 1 contains a rank-1 projection.…”

Section: David Hartleymentioning

confidence: 99%

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2015

Pacific J. Math.

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We establish the existence of Demazure flags for graded local Weyl modules for hyper current algebras in positive characteristic. If the underlying simple Lie algebra is simply laced, the flag has length one; that is, the graded local Weyl modules are isomorphic to Demazure modules. This extends to the positive characteristic setting results of Chari and Loktev, Fourier and Littelmann, and Naoi for current algebras in characteristic zero. Using this result, we prove that the character of local Weyl modules for hyper loop algebras depend only on the highest weight, but not on the (algebraically closed) ground field, and deduce a tensor product factorization for them.

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Section: David Hartleymentioning

confidence: 99%

“…Fix an integer n > 1. Let A D lim k!1 .A k ; k / be the unital simple AH-algebra with tsr A D n constructed by Villadsen [1999]. Then A 1 D C.D n /.…”

Section: David Hartleymentioning

confidence: 99%

Section: David Hartleymentioning

confidence: 99%

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2015

Pacific J. Math.

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“…Note that the class of algebras described in the hypotheses of Theorem 3.11 need not satisfy any of the four equivalent conditions in the conclusion of the same. Indeed, Villadsen has constructed simple, unital AH algebras of arbitrary finite stable rank having unique trace and projections of arbitrarily small trace [32].…”

Section: Theorem 311 Let a Be A Simple Unital And Infinite-dimensimentioning

confidence: 99%

-Stable ASH Algebras

Toms¹,

Winter²

2008

Can. j. math.

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Abstract. The Jiang-Su algebra Z has come to prominence in the classification program for nuclear C * -algebras of late, due primarily to the fact that Elliott's classification conjecture in its strongest form predicts that all simple, separable, and nuclear C * -algebras with unperforated K-theory will absorb Z tensorially, i.e., will be Z-stable. There exist counterexamples which suggest that the conjecture will only hold for simple, nuclear, separable and Z-stable C * -algebras. We prove that virtually all classes of nuclear C * -algebras for which the Elliott conjecture has been confirmed so far consist of Z-stable C * -algebras. This follows in large part from the following result, also proved herein: separable and approximately divisible C * -algebras are Z-stable.

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“…A unital abelian C * -algebra consists of continuous functions on a compact Hausdorff space X ; in this case the nuclear dimension recaptures the dimension of X . At the other extreme lie simple C * -algebras, where nuclear dimension divides the exotic examples of [25,54,65,69] from those accessible to K-theoretic classification. Indeed, through the work of generations of researchers ( [19,27,38,48,64,75] building on numerous works going back to [18]), we now have a complete classification of separable, simple, unital C * -algebras of finite nuclear dimension satisfying Rosenberg and Schochet's universal coefficient theorem (UCT) [56].…”

Section: Introductionmentioning

confidence: 99%

Nuclear dimension of simple $$\mathrm {C}^*$$-algebras

et al. 2020

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We compute the nuclear dimension of separable, simple, unital, nuclear, $${\mathcal {Z}}$$ Z -stable $$\mathrm {C}^*$$ C ∗ -algebras. This makes classification accessible from $${\mathcal {Z}}$$ Z -stability and in particular brings large classes of $$\mathrm {C}^*$$ C ∗ -algebras associated to free and minimal actions of amenable groups on finite dimensional spaces within the scope of the Elliott classification programme.

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On the stable rank of simple C*-algebras

Cited by 91 publications

References 18 publications

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-Stable ASH Algebras

Nuclear dimension of simple $$\mathrm {C}^*$$-algebras

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