论文标题
强烈自我吸收的$ c^*$ - 代数和克利福德等级
Bundles of strongly self-absorbing $C^*$-algebras with a Clifford grading
论文作者
论文摘要
我们将先前的结果扩展到广义dixmier-douady理论中，以分级$ c^*$ - 代数，作为对不变的捆绑$ c^*$代数的显式计算的手段。对于强烈的自我吸收$ c^*$ - 代数$ d $和复杂的Clifford代数$ \ Mathbb {C} \ ell_ {n} $，我们表明分类级别的自动化群体的分类空间$ \ mathrm {aut} _ {\ text {gr}}（\ Mathbb {c} \ ell_ {n} \ otimes \ Mathcal \ Mathcal {k} \ otimes d）$ ADD兼容兼容兼容的无限循环空间结构，从而引起了同居理论，从而引起了共同的理论理论。对于$ d $稳定有限的和$ x $的有限CW复合物，我们表明，张量的产品操作定义了分级$ c^*$ - 带有fibers fibers $ c^*$的同构的组结构异态至$ h^0（x，x，\ mathbb {z}/2）\ oplus \ hat {e}^1_ {d}（x）$。此外，我们建立同构$ \ hat {e}^1_ {d}（x）\ cong h^1（x; \ mathbb {z}/2）\ times _ {_ {_ {tw}} e^1_ {d} \ Mathcal {o} _ \ infty}（x）$，其中$ e^1_ {d}（x）$是将本地小捆分类与纤维$ d \ otimes \ otimes \ mathcal {k k} $分类的组。尤其是$ e^1 _ {\ Mathcal {o} _ \ infty}（x）\ cong h^1（x; \ mathbb {z}/2）\ times _ {_ {_ {tw}} e^1 _ {\ mathcal {z）最后两个因素上的乘法与用纤维的束束相似，分别在有限且无限的尺寸Hilbert空间上的划分的紧凑型算子。
We extend our previous results on generalized Dixmier-Douady theory to graded $C^*$-algebras, as means for explicit computations of the invariants arising for bundles of ungraded $C^*$-algebras. For a strongly self-absorbing $C^*$-algebra $D$ and complex Clifford algebras $\mathbb{C}\ell_{n}$ we show that the classifying spaces of the groups of graded automorphisms $\mathrm{Aut}_{\text{gr}}(\mathbb{C}\ell_{n}\otimes \mathcal{K }\otimes D)$ admit compatible infinite loop space structures giving rise to a cohomology theory $\hat{E}^*_D(X)$. For $D$ stably finite and $X$ a finite CW-complex, we show that the tensor product operation defines a group structure on the isomorphism classes of locally trivial bundles of graded $C^*$-algebras with fibers $ \mathbb{C}\ell_{k}\otimes D \otimes \mathcal{K}$ and that this group is isomorphic to $H^0(X,\mathbb{Z}/2)\oplus \hat{E}^1_{D}(X)$. Moreover, we establish isomorphisms $\hat{E}^1_{D}(X)\cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^1_{D}(X)$ and $\hat{E}^1_{D}(X)\cong E^1_{D\otimes \mathcal{O}_\infty}(X)$, where $E^1_{D}(X)$ is the group that classifies the locally trivial bundles with fibers $D\otimes \mathcal{K}$. In particular $E^1_{\mathcal{O}_\infty}(X)\cong H^1(X;\mathbb{Z}/2) \times_{_{tw}} E^1_{\mathcal{Z}}(X)$ where $\mathcal{Z}$ is the Jiang-Su algebra and the multiplication on the last two factors is twisted similarly to the Brauer theory for bundles with fibers the graded compact operators on a finite and respectively infinite dimensional Hilbert space.
