Limit algebras and integer-valued cocycles, revisited

A multiplicative ergodic theorem for von Neumann algebra valued cocycles

The classical Multiplicative Ergodic Theorem (MET) of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution.

Obstructions to lifting cocycles on groupoids and the associated $C^*$-algebras

Given a short exact sequence of locally compact abelian groups $0 \to A \to B \to C \to 0$ and a continuous $C$-valued $1$-cocycle $ϕ$ on a locally compact Hausdorff groupoid $Γ$ we construct a twist of $Γ$ by $A$ that is trivial if and only if $ϕ$ lifts. The cocycle determines a strongly continuous action of $\widehat{C}$ into $\operatorname{Aut} C^*(Γ)$ and we prove that the $C^*$-algebra of the twist is isomorphic to the induced algebra of this action if $Γ$ is amenable. We apply our results to a groupoid determined by a locally finite cover of a space $X$ and a cocycle provided by a Čech 1…

On the K-theory of twisted higher-rank-graph C*-algebras

We investigate the K-theory of twisted higher-rank-graph algebras by adapting parts of Elliott's computation of the K-theory of the rotation algebras. We show that each 2-cocycle on a higher-rank graph taking values in an abelian group determines a continuous bundle of twisted higher-rank graph algebras over the dual group. We use this to show that for a circle-valued 2-cocycle on a higher-rank graph obtained by exponentiating a real-valued cocycle, the K-theory of the twisted higher-rank graph algebra coincides with that of the untwisted one.

李群值上循环的流形与离散上同调

Manifolds of Lie-Group-Valued Cocycles and Discrete Cohomology

本文证明紧群作用下的李群值相对连续上循环空间构成解析子流形，并分解为纤维丛，由此得出上同调空间离散，并应用于态射空间与射影表示空间的流形结构。

Product-system models for twisted $C^*$-algebras of topological higher-rank graphs

We use product systems of $C^*$-correspondences to introduce twisted $C^*$-algebras of topological higher-rank graphs. We define the notion of a continuous $\mathbb{T}$-valued $2$-cocycle on a topological higher-rank graph, and present examples of such cocycles on large classes of topological higher-rank graphs. To every proper, source-free topological higher-rank graph $Λ$, and continuous $\mathbb{T}$-valued $2$-cocycle $c$ on $Λ$, we associate a product system $X$ of $C_0(Λ^0)$-correspondences built from finite paths in $Λ$. We define the twisted Cuntz--Krieger algebra $C^*(Λ,c)$ to be the C…

通过Borel上循环的算子代数形变

Deformation of operator algebras by Borel cocycles

本文受Kasprzak的Rieffel形变方法启发，利用局部紧群的Borel 2-上循环定义C*-代数的形变，证明其与交叉积的同构关系，并得出满足Baum-Connes猜想的群上该形变的K-理论具有同伦不变性的结论。