On reduced twisted group C*-algebras that are simple and/or have a unique trace

Abstract

We study the problem of determining when the reduced twisted group C *algebra associated with a discrete group G is simple and/or has a unique tracial state, and present new sufficient conditions for this to hold. One of our main tools is a combinatorial property, that we call the relative Kleppner condition, which ensures that a quotient group G/H acts by freely acting automorphisms on the twisted group von Neumann algebra associated to a normal subgroup H. We apply our results to different types of groups, e.g. wreath products and Baumslag-Solitar groups.

Fourier theory andC∗-algebras

“…One may of course also study simplicity and uniqueness of the trace for twisted reduced group C * -algebras, and quite a lot is known. We refer to [84,85] for an overview on this theme, as well as some new results.…”

Smooth lattice orbits of nilpotent groups and strict comparison of projections

“…Hence this condition is necessary for C * r (G, σ) to be simple (i.e., to have no non-trivial ideals), but it is not always sufficient, cf. [11]. See also [87,91] for other results relying on this condition.…”

“…Our main result concerns the existence of frames and Riesz sequences generated by smooth vectors, i.e., vectors g ∈ H π for which the orbit maps x → π(x)g are smooth; in notation, g ∈ H ∞ π . The result relies on a compatibility condition between the 2cocycle σ of the projective representation π and the lattice Γ, known as "Kleppner's condition"; see [11,69,87,88,91]. A pair (Γ, σ) satisfies Kleppner's condition if, for any non-trivial γ ∈ Γ satisfying σ(γ, γ ′ ) = σ(γ ′ , γ) for all γ ′ ∈ Γ such that γ ′ γ = γγ ′ , the associated conjugacy class {(γ ′ ) −1 γγ ′ : γ ′ ∈ Γ} is infinite.…”

Gabor duality theory for Morita equivalent C∗-algebras

“…is a bounded operator, we say g is an (n, d)-matrix Gabor Bessel vector for L 2 (G) with respect to Λ, or that G(g; Λ) is an (n, d)-matrix Gabor Bessel system for L 2 (G). Equivalently, there is D > 0 such that for all f ∈ L 2 (G×Z 6) which may also be written as…”