论文标题
圆圈动作在UHF-Elgebras上的高度高dixmier-douady理论
Equivariant higher Dixmier-Douady Theory for circle actions on UHF-algebras
论文作者
论文摘要
我们为本地琐碎的捆绑图开发了一个dixmier-douady理论，该理论是$ c^*$ - 带有光纤的代数$ d \ otimes \ otimes \ mathbb {k} $，配备了fibrewise $ \ mathbb {t} $ - an \ operatotorname {end} \ left（v \ right）^{\ otimes \ infty} $对于$ \ mathbb {t} $ - 表示$ v $。特别是，我们表明$ \ mathbb {t} $ - equivariant $*$ - 自动形态$ \ perperatorName {aut} _ {\ mathbb {t}}}（d \ otimes \ otimes \ mathbb {k}） $ e^*_ {d，\ Mathbb {t}}（x）$。然后，等效捆绑包的同构类别形成一个组相对于纤维张量张量的产物，该产品是同构对$ e^1_ {d，\ Mathbb {t}}}（x）\ cong [x，x，b \ opperatOrnAme {aut} {aut} {aut} _ {\ mathbb {\ mathbb {\ mathbb {\ t} $ {我们计算用于Tori的组，并将情况比较$ d = \ Mathbb {C} $与Equivariant Brauer组在基本空间上进行琐碎的动作。
We develop an equivariant Dixmier-Douady theory for locally trivial bundles of $C^*$-algebras with fibre $D \otimes \mathbb{K}$ equipped with a fibrewise $\mathbb{T}$-action, where $\mathbb{T}$ denotes the circle group and $D = \operatorname{End}\left(V\right)^{\otimes \infty}$ for a $\mathbb{T}$-representation $V$. In particular, we show that the group of $\mathbb{T}$-equivariant $*$-automorphisms $\operatorname{Aut}_{\mathbb{T}}(D \otimes \mathbb{K})$ is an infinite loop space giving rise to a cohomology theory $E^*_{D,\mathbb{T}}(X)$. Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to $E^1_{D,\mathbb{T}}(X) \cong [X, B\operatorname{Aut}_{\mathbb{T}}(D \otimes \mathbb{K})]$. We compute this group for tori and compare the case $D = \mathbb{C}$ to the equivariant Brauer group for trivial actions on the base space.
