Some computations of 1-cohomology groups and construction of non orbit equivalent actions

Cohomology groups, continuous full groups and continuous orbit equivalence of topological Markov shifts

We will study several subgroups of continuous full groups of one-sided topological Markov shifts from the view points of cohomology groups of full group actions on the shift spaces. We also study continuous orbit equivalence and strongly continuous orbit equivalence in terms of these subgroups of the continuous full groups and the cohomology groups.

Vanishing of the cyclic cohomology of infinite von Neumann algebras

We prove that if A is an infinite von Neumann algebra (i. e., the identity can be decomposed as a sum of a sequence of pairwise disjoint projections, all equivalent to the identity) then the cyclic cohomology of A vanishes. We show that the method of the proof applies to certain algebras of infinite matrices.

Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

For continuously orbit equivalent one-sided topological Markov shifts $(X_A,σ_A)$ and $(X_B,σ_B)$, their eventually periodic points and cocycle functions are studied. As a result we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^A, \bar{H}^A_+)$ and $(\bar{H}^B, \bar{H}^B_+)$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of the ordered cohomology, so that the the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invar…

Localized cohomology and some applications of Popa's cocycle super-rigidity theorem

We prove that orbit equivalence of measure preserving ergodic a.e. free actions of a countable group with the relative property (T) is a complete analytic equivalence relation.

重新审视Galois上同调

Galois cohomology revisited

本文用与簇相关的$C^*$-代数的K-理论重新表述Galois上同调，证明了代数簇的同构等价于对应$C^*$-代数的同构或Morita等价，并以有理椭圆曲线为例进行了详细分析。