Completely positive maps of order zero

We say a completely positive contractive map between two C*-algebras has order zero, if it sends orthogonal elements to orthogonal elements. We prove a structure theorem for such maps. As a consequence, order zero maps are in one-to-one correspondence with *-homomorphisms from the cone over the domain into the target algebra. Moreover, we conclude that tensor products of order zero maps are again order zero, that the composition of an order zero map with a tracial functional is again a tracial functional, and that order zero maps respect the Cuntz relation, hence induce ordered semigroup morphisms between Cuntz semigroups.Comment: 13 page

The Stable Rank of Diagonal ASH Algebras and Crossed Products by Minimal Homeomorphisms

We introduce a subclass of recursive subhomogeneous algebras, in which each of the pullback maps is diagonal in a suitable sense. We define the notion of a diagonal map between two such algebras and show that every simple inductive limit of these algebras with diagonal maps has stable rank one. As an application, we prove that for any infinite compact metric space $T$ and minimal homeomorphism $h\colon T\to T$, the associated dynamical crossed product $\mathrm{C^*}(\mathbb{Z},T,h)$ has stable rank one. This affirms a conjecture of Archey, Niu, and Phillips. We also show that the Toms-Winter Conjecture holds for such crossed products.Comment: In v2, a few of the proofs have been shortened, some figures have been improved, and another crossed product application has been added. Some of the sections and results have been reorganized to aid readability. A few typos have been fixed and minor corrections have been mad

The universal functorial equivariant Lefschetz invariant

We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K_0 of the category of "phi-endomorphisms of finitely generated free RPi(G,X)-modules". We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.Comment: 33 pages; shortened version of the author's PhD thesis, supervised by Wolfgang Lueck, Westfaelische Wilhelms-Universitaet Muenster, 200

On the Relationship between Normative Claims and Empirical Realities in Immigration

What is and what ought to be the relationship between empirical research and normative analysis with respect to migration policies? The paper addresses this question from the perspective of political theory, asking about the place of empirical research in philosophical discussions of migration, and, for the most part, leaving to others questions about what role, if any, normative considerations do and should play in empirical research on migration. At the outset the paper also takes note of one important way in which empirical research can and should contribute to normative discussions of migration, quite apart from its role in contributing to political philosophy. DOI: 10.17879/15199614880

Fixed points for actions of Aut(Fn) on CAT(0) spaces

For n greater or equal 4 we discuss questions concerning global fixed points for isometric actions of Aut(Fn), the automorphism group of a free group of rank n, on complete CAT(0) spaces. We prove that whenever Aut(Fn) acts by isometries on complete d-dimensional CAT(0) space with d is less than 2 times the integer function of n over 4 and minus 1, then it must fix a point. This property has implications for irreducible representations of Aut(Fn), which are also presented here. For SAut(Fn), the unique subgroup of index two in Aut(Fn), we obtain similar results

Introduction: Why Should We Study Migration Policies at the Interface between Empirical Research and Normative Analysis?

The text introduces the concept behind the Proceedings of the 2018 ZiF Workshop “Studying Migration Policies at the Interface between Empirical Research and Normative Analysis”. It explains why there is a need to study migration policies across disciplines, includes a short note on the current literature, and provides a look back at the workshop. DOI:10.17879/1519962468

Graph Laplacians, component groups and Drinfeld modular curves

Let $\frak{p}$ be a prime ideal of $\mathbb{F}_q[T]$. Let $J_0(\frak{p})$ be the Jacobian variety of the Drinfeld modular curve $X_0(\frak{p})$. Let $\Phi$ be the component group of $J_0(\frak{p})$ at the place $1/T$. We use graph Laplacians to estimate the order of $\Phi$ as $\mathrm{deg}(\frak{p})$ goes to infinity. This estimate implies that $\Phi$ is not annihilated by the Eisenstein ideal of the Hecke algebra $\mathbb{T}(\frak{p})$ acting on $J_0(\frak{p})$ once the degree of $\frak{p}$ is large enough. We also obtain an asymptotic formula for the size of the discriminant of $\mathbb{T}(\frak{p})$ by relating this discriminant to the order of $\Phi$; in this problem the order of $\Phi$ plays a role similar to the Faltings height of classical modular Jacobians. Finally, we bound the spectrum of the adjacency operator of a finite subgraph of an infinite diagram in terms of the spectrum of the adjacency operator of the diagram itself; this result has applications to the gonality of Drinfeld modular curves

Splitting Polytopes

A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to the splits of $P$ (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope $P$, the split complex of $P$. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change

Kirchberg-Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case

We show that in order to prove that all locally compact groups with exact reduced group C∗-algebras are exact in the dynamical sense (i.e., KW-exact), it suffices to show this for totally disconnected locally compact groups

Mahler measures and Fuglede--Kadison determinants

The Mahler measure of a function on the real d-torus is its geometric mean over the torus. It appears in number theory, ergodic theory and other fields. The Fuglede-Kadison determinant is defined in the context of von Neumann algebra theory and can be seen as a noncommutative generalization of the Mahler measure. In the paper we discuss and compare theorems in both fields, especially approximation theorems by finite dimensional determinants. We also explain how to view Fuglede-Kadison determinants as continuous functions on the space of marked groups