Quasidiagonal representations of nilpotent groups

Abstract

We show that every unitary representation of a discrete solvable virtually nilpotent group G is quasidiagonal. Roughly speaking, this says that every unitary representation of G approximately decomposes as a direct sum of finite dimensional approximate representations. In operator algebraic terms we show that C * (G) is strongly quasidiagonal.

Finite decomposition rank for virtually nilpotent groups

“…Hence C * π (G) must appear as a direct summand of C * π φ (G). Moreover, [7] proves that M G/N ⊗ C * (N) is strongly quasidiagonal whenever N is nilpotent. Since C * π φ (G) embeds into M G/N ⊗ C * πτ (N), the C*-algebra C * π φ (G) is quasidiagonal and therefore so is C * π (G).…”

“…We prove Equation (6.1) by the same method as [7,Lemma 3.3]. For each z ∈ C * (G) and each x ∈ F , there is a unique z x ∈ C * (N) such that…”

“…By Theorem 2.3, the C*-algebra C * π (G) is simple and has a unique trace. By [7] it is quasidiagonal and by [11] it has finite nuclear dimension. Therefore by Theorem 2.8 we obtain dr(C * π (G)) ≤ 1.…”

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

“…Since B θ is the C*-algebra generated by an irreducible representation of a finitely generated nilpotent group, it is simple with a unique trace (this is well known, see e.g. the introduction of [5]). It follows from [6, Theorems 2.9 & 4.4 ] that B θ has strict comparison.…”

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