Finite decomposition rank for virtually nilpotent groups

Abstract

We show that inductive limits of virtually nilpotent groups have strongly quasidiagonal C*-algebras, extending results of the first author on solvable virtually nilpotent groups. We use this result to show that the decomposition rank of the group C*-algebra of a finitely generated virtually nilpotent group G is bounded by 2 · h(G)! − 1, where h(G) is the Hirsch length of G. This extends and sharpens results of the first and third authors on finitely generated nilpotent groups. It then follows that if a C*-algebra generated by an irreducible representation of a virtually nilpotent group satisfies the universal coefficient theorem, it is classified by its Elliott invariant.

Smooth lattice orbits of nilpotent groups and strict comparison of projections

“…Twisted group C*-algebras of finitely generated, nilpotent groups. In this section we will extend the results from [32,33] on the finite decomposition rank (nuclear dimension) of group C * -algebras associated to finitely generated nilpotent groups to twisted group algebras of such groups. By Theorem 5.1, this will imply the presence of strict comparison of projections, which will allow us to exploit Proposition 5.2 in the setting of Section 4.…”

“…Finite decomposition rank of group C * -algebras associated to finitely generated, nilpotent groups is due to Eckhardt, Gillaspy and McKenney [32,33]. Our proof of Theorem 1.6 relies on [32] and extends their result to the twisted case via the theory of representation groups; in particular, we use a recent result of Hatui, Narayanan and Singla [57, Theorem 3.5], cf. Section 5.3 for precise details.…”

“…Extensions. The proof of Theorem 1.3 makes a fundamental use of the presence of strict comparison of projections in twisted group C * -algebras C * (Γ, σ) of finitely generated nilpotent groups Γ, which we show by relying on the paper [32]. The results in [32] are, however, also valid for virtually nilpotent groups, and it might be that a version of Theorem 1.3 is also valid for more general (classes of) groups of polynomial growth.…”

“…The proof of Theorem 1.3 makes a fundamental use of the presence of strict comparison of projections in twisted group C * -algebras C * (Γ, σ) of finitely generated nilpotent groups Γ, which we show by relying on the paper [32]. The results in [32] are, however, also valid for virtually nilpotent groups, and it might be that a version of Theorem 1.3 is also valid for more general (classes of) groups of polynomial growth. For such an extension, several parts in our approach, among others, the explicitly constructed Hilbert C * -modules (Theorem 1.4) would require different arguments, while there are also several ingredients that do currently not have analogues for virtual nilpotent groups, among others, the existence of suitable representation groups [57].…”

Faithful tracial states on quotients of C*-algebras

“…In [14] it is proven that it holds true for (primitive) quotients of C * (G), whenever G is finitely generated and nilpotent. However, it is seemingly very difficult to extend these results, and the authors of [15] argue that resolving it for virtually nilpotent groups might not be any easier than resolving the UCT-problem for nuclear C * -algebras. If all nuclear C * -algebras were shown to be in the UCT-class, then the assumption of all quotients satisfying the UCT would, of course, be vacuous in Theorem 3.24 for nuclear C * -algebras, but [16, Proposition 5.1] would also resolve this.…”