Quasidiagonal Representations of Nilpotent Groups

TL;DR

This paper proves that all unitary representations of solvable virtually nilpotent groups are quasidiagonal, implying their associated C*-algebras are strongly quasidiagonal, which advances understanding of their operator algebraic structure.

Contribution

It establishes that every unitary representation of such groups is quasidiagonal and that their C*-algebras are strongly quasidiagonal, a novel result in operator algebra theory.

Findings

All unitary representations of G are quasidiagonal.

C*(G) is strongly quasidiagonal.

Representation decompositions are approximately finite dimensional.

Abstract

We show that every unitary representation of a solvable discrete 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.