关于闭算子凸锥的吸收表示
Absorbing representations with respect to closed operator convex cones
摘要：
首次研究了$C^\ast$ -代数关于闭算子凸锥的吸收表示.我们完全确定当这种吸收表示存在，这导致了问题的特点时，表示吸收，在经典的魏尔-冯诺依曼型定理Voiculescu。在经典的情况下，这是由艾略特和Kucerovsky证明，他们证明了一个表示是核吸收的，当且仅当它引起一个纯粹的大扩展。通过考虑$C^\ast$ -代数的扩展的相关问题，我们称之为纯粹的大问题，我们问当一个纯粹的largeness条件类似于一个定义的艾略特和Kucerovsky，意味着吸收关于一些给定的闭算子凸锥。我们解决了由有限拓扑空间在$C^\ast$ -代数上的作用诱导的一类特殊的闭算子凸锥的问题。作为这一结果的应用，我们给出了某些含有纯无限的，双边的，闭理想的商是AF代数的$C^\ast$ -代数的$K$ -理论分类。这推广了第二作者S. Eilers和G.在Restorff中，所有扩展都必须是满的。
英文摘要：
We initiate the study of absorbing representations of $C^\ast$-algebras with respect to closed operator convex cones. We completely determine when such absorbing representations exist, which leads to the question of characterising when a representation is absorbing, as in the classical Weyl-von Neumann type theorem of Voiculescu. In the classical case, this was proven by Elliott and Kucerovsky who proved that a representation is nuclearly absorbing if and only if it induces a purely large extension. By considering a related problem for extensions of $C^\ast$-algebras, which we call the purely large problem, we ask when a purely largeness condition similar to the one defined by Elliott and Kucerovsky, implies absorption with respect to some given closed operator convex cone. We solve this question for a special type of closed operator convex cone induced by actions of finite topological spaces on $C^\ast$-algebras. As an application of this result, we give $K$-theoretic classification for certain $C^\ast$-algebras containing a purely infinite, two-sided, closed ideal for which the quotient is an AF algebra. This generalises a similar result by the second author, S. Eilers and G. Restorff in which all extensions had to be full.
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