Finitely generated nilpotent group C*-algebras have finite nuclear dimension

It was recently shown that each C*-algebra generated by a faithful irreducible representation of a finitely generated, torsion free nilpotent group is classified by its ordered K-theory. For the three step nilpotent group U T (4, Z) we calculate the ordered K-theory of each C*-algebra generated by a faithful irreducible representation of U T (4, Z) and see that they are all simple AT algebras. We also point out that there are many simple non AT algebras generated by irreducible representations of nilpotent groups.

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“…Theorem 2.3 (See [5,6]). Let Γ be a torsion free finitely generated nilpotent group and π a faithful irreducible representation of Γ.…”

Section: Preliminariesmentioning

confidence: 99%

“…Also by [29], it follows that (A θ ⊗ A θ ) ⋊ β Z satisfies the universal coefficient theorem. By [5], B θ is quasidiagonal and by [6] B θ has finite nuclear dimension. By Theorem 2.1, B θ is therefore isomorphic to an AT algebra.…”

Section: Elliott Invariantsmentioning

confidence: 99%

Classification of C*-algebras generated by representations of the unitriangular group UT(4,Z)

Eckhardt

Kleski

McKenney

2016

Journal of Functional Analysis

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“…Therefore our job consists of showing that C * π (G) ∈ C. We have known for a while that C * π (G) is nuclear [8] and simple [12]. Recently, the first author together with McKenney [7] showed that C * π (G) has finite nuclear dimension (this work directly relied on a long list of results including [5,10,11,14,18] -see the introduction of [7] for the full story). As pointed out in [7,Theorem 4.5] the only missing ingredient to show C * π (G) ∈ C was the UCT.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the first author together with McKenney [7] showed that C * π (G) has finite nuclear dimension (this work directly relied on a long list of results including [5,10,11,14,18] -see the introduction of [7] for the full story). As pointed out in [7,Theorem 4.5] the only missing ingredient to show C * π (G) ∈ C was the UCT. It was also pointed out in [7,Theorem 4.6] that if G is torsion free and π is a faithful representation then C * π (G) satisfies the UCT.…”

Section: Introductionmentioning

confidence: 99%

“…As pointed out in [7,Theorem 4.5] the only missing ingredient to show C * π (G) ∈ C was the UCT. It was also pointed out in [7,Theorem 4.6] that if G is torsion free and π is a faithful representation then C * π (G) satisfies the UCT. This observation has already found success in [6] where the authors calculated the Elliott invariant of C*-algebras generated by faithful irreducible representations of the (torsion free) unitriangular group U T (4, Z), thus classifying them within C.…”

Section: Introductionmentioning

confidence: 99%

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Irreducible representations of nilpotent groups generate classifiable C*-algebras

Eckhardt¹,

Gillaspy²

2015

Preprint

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We show that C*-algebras generated by irreducible representations of finitely generated nilpotent groups satisfy the universal coefficient theorem of Rosenberg and Schochet. This result combines with previous work to show that these algebras are classifiable by their Elliott invariants within the class of unital, simple, separable, nuclear C*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. We also show that these C*-algebras are central cutdowns of twisted group C*-algebras with homotopically trivial cocycles.

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The nuclear dimension of C⁎-algebras associated to homeomorphisms

Hirshberg

2017

Advances in Mathematics

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We show that if X is a finite dimensional locally compact Hausdorff space, then the crossed product of C0(X) by any automorphism has finite nuclear dimension. This generalizes previous results, in which the automorphism was required to be free. As an application, we show that group C * -algebras of certain non-nilpotent groups have finite nuclear dimension.Nuclear dimension for C * -algebras was introduced by Winter and Zacharias in [WZ10], as a noncommutative generalization of covering dimension. This is a variant of the previous notion of decomposition rank ([KW04]), and is also applicable to non-quasidiagonal C * -algebras. Since then, it has come to play a major role in structure and classification of C * -algebras. It was shown in [WZ10] that if X is a locally compact metrizable space, then dim nuc (C 0 (X)) coincides with the covering dimension of X, and the property of having finite nuclear dimension is preserved under various constructions: forming direct sums and tensor products, passing to quotients and hereditary subalgebras, and forming extensions. An important problem which was left open in [WZ10] is to understand the behavior of finite nuclear dimension under forming crossed products. It was shown in [TW13] that finite nuclear dimension passes to crossed products by minimal homeomorphisms: if X is a compact metric space with finite covering dimension and h : X → X is a minimal homeomorphism, then denoting α(f ) = f • h, we have dim nuc (C(X) ⋊ α Z) < ∞. This was re-proved in a different way in [HWZ15]. The paper [HWZ15] develops a notion of Rokhlin dimension for an automorphism of a C * -algebra (extended in [HP15] to the non-unital setting). It was shown there that in general, if A has finite nuclear dimension and α ∈ Aut(A) has finite Rokhlin dimension, then A ⋊ α Z has finite nuclear dimension as well, and furthermore, for a minimal homeomorphism as above, the induced automorphism on C(X) always has finite Rokhlin dimension. Szabó ([Sza15b]) then showed that the minimality condition can be weakened to freeness: if X is as above and h : X → X has no periodic points, then α has finite Rokhlin dimension (and therefore, by [HWZ15, Theorem 4.1], the crossed product has finite nuclear dimension). In fact, Szabó's result works for actions of Z m as well. This uses the marker property, introduced by Gutman in [Gut15b]. Those results were further extended to free actions of finitely generated nilpotent groups in [SWZ14].For the case of integer actions arising from homeomorphisms, this leaves the case of actions which also have periodic points. Those include important examples. For instance, suppose G is a countable abelian group and G has finite covering dimension, and suppose α is an automorphism of G. The group C * -algebra C * (G⋊ α Z) is isomorphic to a crossed product C( G) ⋊ Z, and such actions are never free: an

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Finitely generated nilpotent group C*-algebras have finite nuclear dimension

Cited by 4 publications

References 0 publications

Classification of C*-algebras generated by representations of the unitriangular group UT(4,Z)

Classification of C*-algebras generated by representations of the unitriangular group UT(4,Z)

Irreducible representations of nilpotent groups generate classifiable C*-algebras

The nuclear dimension of C⁎-algebras associated to homeomorphisms

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