English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

To study outer actions $\alpha$ of a group $G$ on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$, $0<\lambda<1$, we study first the cohomology group of a group with the unitary group of an abelian von Neumann algebra as a coefficient group and establish a technique to reduce the coefficient group to the torus $\mathsf T$ by the Shapiro mechanism based on the groupoid approach. We then show a functorial construction of outer actions of a countable discrete amenable group on an AFD factor of type $\mathrm{III}_\lambda$, sharpening the result in [17, §4]. The periodicity of the flow of weights on a factor $\mathcal M$ of type $\mathrm{III}_\lambda$ allows us to introduce an equivariant commutative square directly related to the discrete core. But this makes it necessary to introduce an enlarged group $\mathrm{Aut}(\mathcal M)_{m}$ relative to the modulus homomorphism $m=\mod\colon \mathrm{Aut}(\mathcal M)\to \mathsf R/T'\mathsf Z$. We then discuss the reduced modified HJR-exact sequence, which allows us to describe the invariant of outer action $\alpha$ in a simpler form than the one for a general AFD factor: for example, the cohomology group $H_{m,s}^{out}(G,N,\mathsf T)$ of modular obstructions is a compact abelian group. Making use of these reductions, we prove the classification result of outer actions of $G$ on an AFD factor $\mathcal M$ of type $\mathrm{III}_{\lambda}$.

Outer actions of a discrete amenable group on approximately finite dimensional factors II, the III$_{\lambda}$-case, $\lambda\neq 0$