论文标题
广义的渐近代数和$ \ mathrm {e} $ - 不可分离的$ \ mathrm {c}^*$ - 代数的理论
Generalized asymptotic algebras and $\mathrm{E}$-theory for non-separable $\mathrm{C}^*$-algebras
论文作者
论文摘要
在$ \ mathrm {e} $的先前定义中 - 理论，需要$ \ mathrm {c}^*$ - 代数的可分离性来构造构图产品或证明长序列。考虑到后者，可以追溯到这些$ \ mathrm {e} $ - 理论组可容纳有关一个真实参数输入无穷大的信息的可能性，但不是针对有指示集对渐近的渐近性，可以追溯到以下事实。我们提出了$ \ mathrm {e} $ - 理论的定义，该定义还通过概括了渐近代数的概念来结合这些其他信息。结果，它不仅具有所有期望的产品，而且还具有所有长的精确序列，即使对于不可分割的$ \ mathrm {c}^*$ - 代数。更确切地说，我们的构造产生了均值$ \ mathrm {e} $ - $ \ mathbb {z} _2 $ -graded $ g $ - $ g $ - $ \ mathrm {c}^*$ - 任意离散$ g $的代数。我们怀疑我们的$ \ mathrm {e} $的模型 - 理论可能是研究无限尺寸流形的索引理论的正确实体。
In previous definition of $\mathrm{E}$-theory, separability of the $\mathrm{C}^*$-algebras is needed either to construct the composition product or to prove the long exact sequences. Considering the latter, the potential failure of the long exact sequences can be traced back to the fact that these $\mathrm{E}$-theory groups accommodate information about asymptotic processes in which one real parameter goes to infinity, but not about more complicated asymptotics parametrized by directed sets. We propose a definition for $\mathrm{E}$-theory which also incorporates this additional information by generalizing the notion of asymptotic algebras. As a consequence, it not only has all desirable products but also all long exact sequences, even for non-separable $\mathrm{C}^*$-algebras. More precisely, our construction yields equivariant $\mathrm{E}$-theory for $\mathbb{Z}_2$-graded $G$-$\mathrm{C}^*$-algebras for arbitrary discrete groups $G$. We suspect that our model for $\mathrm{E}$-theory could be the right entity to investigate index theory on infinite dimensional manifolds.
