On the classification problem for nuclear C<sup>∗</sup>-algebras

Toms, Andrew S.

doi:10.4007/annals.2008.167.1029

Cited by 155 publications

(158 citation statements)

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“…Remarkably, there exist algebras satisfying the hypotheses of the above theorem which are not Z-stable ( [51], [56], [57] We shall see later that for a substantial class of simple, separable, amenable, and stably finite C * -algebras, all three of our regularity properties are equivalent. Moreover, the algebras in this class which do satisfy these three properties also satisfy (EC).…”

Section: Regularity Propertiesmentioning

confidence: 99%

“…Second, the Cuntz semigroup unifies the counterexamples of Rørdam and the second-named author. One can show that the examples of [50], [56], and [57] all consist of pairs of algebras with different Cuntz semigroups; there are no known counterexamples to the conjecture that simple separable amenable C * -algebras will be classified up to * -isomorphism by the Elliott invariant and the Cuntz semigroup.…”

Section: Regularity Properties For Amenable C * -Algebras 241mentioning

confidence: 99%

“…In [57], the second-named author constructed a pair of simple unital AH algebras which, while non-isomorphic, agreed on a wide swath of invariants including the Elliott invariant, all continuous (with respect to inductive sequences) and homotopy invariant functors from the category of C * -algebras (a class which includes the Murray-von Neumann semigroup), the real and stable ranks, and, as was shown later in [58], stable isomorphism invariants (those invariants which are insensitive to tensoring with a matrix algebra or passing to a hereditary subalgebra). It seemed reasonable to expect that the distinguishing invariant in this example-the Cuntz semigroup-might be a good candidate for enlarging the invariant.…”

Section: Regularity Properties For Amenable C * -Algebras 241mentioning

confidence: 99%

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Regularity properties in the classification program for separable amenable C*-algebras

Elliott

Toms

2008

Bull. Amer. Math. Soc.

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145

137

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Abstract. We report on recent progress in the program to classify separable amenable C * -algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.

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Section: Regularity Propertiesmentioning

confidence: 99%

Section: Regularity Properties For Amenable C * -Algebras 241mentioning

confidence: 99%

Section: Regularity Properties For Amenable C * -Algebras 241mentioning

confidence: 99%

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Regularity properties in the classification program for separable amenable C*-algebras

Elliott

Toms

2008

Bull. Amer. Math. Soc.

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145

137

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“…[2] and Theorem 2.5), but the result can't be extended to all AH algebras. Indeed, the reader will find in [16] a pair of positive elements in a simple unital AH algebra of stable rank one such that the corresponding Hilbert modules, say E and F , are not isomorphic but do satisfyP E =P F .…”

Section: Classifying Hilbert Modulesmentioning

confidence: 99%

Three Applications of the Cuntz Semigroup

Brown

Toms

2007

International Mathematics Research Notices

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Abstract. Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup:(i) for many simple C * -algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott's classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; (iii) for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant.

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“…In light of the fact that there exist such C * -algebras that cannot be classified by ordered K-theory and traces [34], the current trend is to try to identify, using regularity properties, the C * -algebras which are (or should be) classifiable. This idea is crystallized in a conjecture (cf.…”

Section: Introductionmentioning

confidence: 99%

Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

Tikuisis

2013

Math. Ann.

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Abstract. The main result here is that a simple separable C * -algebra is Z-stable (where Z denotes the Jiang-Su algebra) if (i) it has finite nuclear dimension or (ii) it is approximately subhomogeneous with slow dimension growth. This generalizes the main results of [33,42] to the nonunital setting. As a consequence, finite nuclear dimension implies Z-stability even in the case of a separable C * -algebra with finitely many ideals. Algebraic simplicity is established as a fruitful weakening of being simple and unital, and the proof of the main result makes heavy use of this concept. IntroductionThe program to classify unital, simple, separable, nuclear C * -algebras has recently seen a small paradigm shift [13]. In light of the fact that there exist such C * -algebras that cannot be classified by ordered K-theory and traces [34], the current trend is to try to identify, using regularity properties, the C * -algebras which are (or should be) classifiable. This idea is crystallized in a conjecture (cf. [36, Section 3.5]), due to Andrew Toms and Wilhelm Winter, that among these C * -algebras, the following properties are equivalent:(i) finite nuclear dimension (defined in [44]); (ii) Z-stability (being isomorphic to one's tensor product with the Jiang-Su algebra Z, introduced in [18]); and (iii) almost unperforated Cuntz semigroup. One might also add that, modulo the UCT, conditions (i)-(iii) should be equivalent to being classifiable (though in this form, such a statement is not well-formed because classifiability is a property of a class of C * -algebras, and not a property of a single C * -algebra). At the same time, certain examples of simple, stably projectionless C * -algebras have emerged -including certain crossed products of O 2 by R [21], a nuclear, separable, non-Z-stable example [31, Theorem 4.1], and others [12,17,37]. Certain tools, old and new, already allow one to understand some of the structure of these algebras under special hypotheses [6,27]. However, most of the theory on regularity properties for C * -algebras was developed only in the unital case, and leads one to ask what obstructions (if any) exist with nonunital algebras. Certainly, the unital case carries with it a number of simplifications: for instance, the simplex of traces on a C * -algebra which take the value 1 at the unit (i.e. which are states) plays an indispensible role in the theory, and it is far from obvious what the correct replacement is in the nonunital case. Nonetheless, one hopes that the simplifications in the unital case are superficial, and that ultimately, one can find and prove analogues or generalizations of the known unital results.2010 Mathematics Subject Classification. 46L35, 46L80, 46L05, 47L40, 46L85.

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On the classification problem for nuclear C^∗-algebras

Cited by 155 publications

References 12 publications

Regularity properties in the classification program for separable amenable C*-algebras

Regularity properties in the classification program for separable amenable C*-algebras

Three Applications of the Cuntz Semigroup

Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

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On the classification problem for nuclear C∗-algebras

Cited by 155 publications

References 12 publications

Regularity properties in the classification program for separable amenable C*-algebras

Regularity properties in the classification program for separable amenable C*-algebras

Three Applications of the Cuntz Semigroup

Nuclear dimension, $$\mathcal{Z }$$ Z -stability, and algebraic simplicity for stably projectionless $$C^*$$ C ∗ -algebras

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On the classification problem for nuclear C^∗-algebras