AF-embeddings into C∗-algebras of real rank zero

Perera, Francesc; Rørdam, Mikael

doi:10.1016/j.jfa.2004.05.001

Cited by 34 publications

(36 citation statements)

References 27 publications

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“…Then, [30] and the references therein) and it is Z-stable by [37]. If A has real rank zero, it is weakly divisible by [26]. The next observation says that all these properties coincide for strongly self-absorbing C * -algebras with projections and, in the purely infinite case, are automatically fulfilled.…”

Section: Andrew S Toms and Wilhelm Wintermentioning

confidence: 87%

Strongly self-absorbing $C^{*}$-algebras

Toms

Winter

2007

Trans. Amer. Math. Soc.

239

312

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Abstract. Say that a separable, unital C * -algebra D C is strongly selfabsorbing if there exists an isomorphism ϕ : D → D ⊗ D such that ϕ and id D ⊗ 1 D are approximately unitarily equivalent * -homomorphisms. We study this class of algebras, which includes the Cuntz algebras O 2 , O ∞ , the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of O ∞ with UHF algebras of infinite type. Given a strongly self-absorbing C * -algebra D we characterise when a separable C * -algebra absorbs D tensorially (i.e., is D-stable), and prove closure properties for the class of separable D-stable C * -algebras. Finally, we compute the possible K-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C * -algebras.

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Section: Andrew S Toms and Wilhelm Wintermentioning

confidence: 87%

Strongly self-absorbing $C^{*}$-algebras

Toms

Winter

2007

Trans. Amer. Math. Soc.

239

312

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“…The purpose of this section is to consider a weak form of divisibility that appears quite frequently both in ring theory and operator algebras (see [14] and [3]). In the sequel it will be important to apply this to the set of full projections, which in the monoid-theoretical context corresponds to the set of order units.…”

Section: Weak Divisibilitymentioning

confidence: 99%

“…Observe that if M is a refinement monoid, then u is an atom in M if and only if u is an ideal of M and u is isomorphic to Z Remark 2.8. In the countable case, which will be of interest as V(A) is countable whenever A is a separable C * -algebra, Proposition 2.6 can be obtained by the arguments in [14]. We briefly indicate how to proceed in that case.…”

Section: Weak Divisibilitymentioning

confidence: 99%

“…First, recall that a dimension monoid is, by definition, an inductive limit of simplicial monoids or, equivalently, a monoid that can be represented as V(A) for an AF-algebra A. Now, if M is a countable refinement monoid, use [14,Theorem 3.9] to find a (countable) dimension monoid Δ and a surjective map α : Δ → M such that x ∝ y if and only if α(x) ∝ α(y). (Here, x ∝ y means that x ≤ ny for some natural number n.) In particular, such a map induces an isomorphism between the ideal lattices of Δ and M .…”

Section: Weak Divisibilitymentioning

confidence: 99%

“…By the observation above, that u is an ideal in M and u ∼ = Z + if and only if u is an atom in M , one can see that the map α also induces a one-to-one correspondence between atoms in Δ and atoms in M . This allows us to reduce the problem to dimension monoids, and in that case the result holds after [14,Proposition 5.6].…”

Section: Weak Divisibilitymentioning

confidence: 99%

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Untitled

2011

Trans. Amer. Math. Soc.

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Abstract. The Corona Factorization Property of a C * -algebra, originally defined to study extensions of C * -algebras, has turned out to say something important about intrinsic structural properties of the C * -algebra. We show in this paper that a σ-unital C * -algebra A of real rank zero has the Corona Factorization Property if and only if its monoid V(A) of Murray-von Neumann equivalence classes of projections in matrix algebras over A has a certain (rather weak) comparability property that we call the Corona Factorization Property (for monoids). We show that a projection in such a C * -algebra is properly infinite if (and only if) a multiple of it is properly infinite.The latter result is obtained from some more general results that we establish about conical refinement monoids. We show that the set of order units (together with the zero-element) in a conical refinement monoid is again a refinement monoid under the assumption that the monoid satisfies weak divisibility; and if u is an element in a refinement monoid such that nu is properly infinite, then u can be written as a sum u = s + t such that ns and nt are properly infinite.

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Perturbation of Hausdorff Moment Sequences, and an Application to the Theory of C*-Algebras of Real Rank Zero

Elliott

Rørdam

2006

Operator Algebras

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We investigate the class of unital C * -algebras that admit a unital embedding into every unital C * -algebra of real rank zero, that has no finite-dimensional quotients. We refer to a C * -algebra in this class as an initial object. We show that there are many initial objects, including for example some unital, simple, infinite-dimensional AFalgebras, the Jiang-Su algebra, and the GICAR-algebra.That the GICAR-algebra is an initial object follows from an analysis of Hausdorff moment sequences. It is shown that a dense set of Hausdorff moment sequences belong to a given dense subgroup of the real numbers, and hence that the Hausdorff moment problem can be solved (in a non-trivial way) when the moments are required to belong to an arbitrary simple dimension group (i.e., unperforated simple ordered group with the Riesz decomposition property).

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AF-embeddings into C∗-algebras of real rank zero

Cited by 34 publications

References 27 publications

Strongly self-absorbing $C^{*}$-algebras

Strongly self-absorbing $C^{*}$-algebras

Untitled

Perturbation of Hausdorff Moment Sequences, and an Application to the Theory of C*-Algebras of Real Rank Zero

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