We examine the question of quasidiagonality for C*-algebras of discrete amenable groups from a variety of angles. We give a quantitative version of Rosenberg's theorem via paradoxical decompositions and a characterization of quasidiagonality for group C*-algebras in terms of embeddability of the groups. We consider several notable examples of groups, such as topological full groups associated with Cantor minimal systems and Abels' celebrated example of a finitely presented solvable group that is not residually finite, and show that they have quasidiagonal C*-algebras. Finally, we study strong quasidiagonality for group C*-algebras, exhibiting classes of amenable groups with and without strongly quasidiagonal C*algebras.
Abstract. Nuclear C * -algebras enjoy a number of approximation properties, most famously the completely positive approximation property. This was recently sharpened to arrange for the incoming maps to be sums of order-zero maps. We show that, in addition, the outgoing maps can be chosen to be asymptotically order-zero. Further these maps can be chosen to be asymptotically multiplicative if and only if the C * -algebra and all its traces are quasidiagonal.
By a quasi-representation of a group G we mean an approximately multiplicative map of G to the unitary group of a unital C * -algebra. A quasirepresentation induces a partially defined map at the level K-theory.In the early 90s Exel and Loring associated two invariants to almost-commuting pairs of unitary matrices u and v: one a K-theoretic invariant, which may be regarded as the image of the Bott element in K0(C(T 2 )) under a map induced by quasi-representation of Z 2 in U (n); the other is the winding number in C \ {0} of the closed path t → det(tvu + (1 − t)uv). The so-called Exel-Loring formula states that these two invariants coincide if uv − vu is sufficiently small.A generalization of the Exel-Loring formula for quasi-representations of a surface group taking values in U (n) was given by the second-named author. Here we further extend this formula for quasi-representations of a surface group taking values in the unitary group of a tracial unital C * -algebra.
We give a rigorous account and prove continuity properties for the correspondence between almost flat bundles on a triangularizable compact connected space and the quasi-representations of its fundamental group. For a discrete countable group Γ with finite classifying space BΓ, we study a correspondence between between almost flat K-theory classes on BΓ and group homomorphism K 0 (C * (Γ)) → Z that are implemented by pairs of discrete asymptotic homomorphisms from C * (Γ) to matrix algebras.
A residually finite group acts on a profinite completion by left translation. We consider the corresponding crossed product C * -algebra for discrete countable groups that are central extensions of finitely generated abelian groups by finitely generated abelian groups (these are automatically residually finite). We prove that all such crossed products are classifiable by K-theoretic invariants using techniques from the classification theory for nuclear C * -algebras.
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