论文标题
半锁定模块和无界卡斯帕罗夫产品的本地化
Localisations of half-closed modules and the unbounded Kasparov product
论文作者
论文摘要
在Kasparov产品的背景下，Kucerovsky的著名定理为无界的Kasparov模块提供了足够的条件，以代表其他两个无界Kasparov模块的（内部）Kasparov产品。在本文中，我们讨论了Kucerovsky定理的几种改进和广泛的变体。首先，我们通过通过相对下限代替下限来提供概括，从而放松阳性条件。其次，我们还在半闭合模块的背景下讨论了库塞罗夫斯基的定理，该定理将无界的卡斯帕罗夫模块概括为对称（而不是自偶会）运算符。为了处理此类非同伴操作员的阳性条件，我们引入了一个相当通用的本地化程序，该程序（使用合适的近似单元）为半锁定模块的KK级提供了“本地化代表”。然后，我们使用此本地化过程，证明了Kucerovsky定理的几种半闭合模块的变体。本地化方法的一个明显优势，也是在自我伴侣操作员的特殊情况下（即对于无限的卡斯帕罗夫模块），是，库卡罗夫斯基原始定理中的（全球）阳性状况被（不太约束性的）“本地”积极条件代替了，在精神上，它在精神上更加接近康涅莱斯 - 康雷斯·萨尔斯·萨尔斯·萨尔斯（Connes-Skandalis-Sares）的精神。
In the context of the Kasparov product in unbounded KK-theory, a well-known theorem by Kucerovsky provides sufficient conditions for an unbounded Kasparov module to represent the (internal) Kasparov product of two other unbounded Kasparov modules. In this article, we discuss several improved and generalised variants of Kucerovsky's theorem. First, we provide a generalisation which relaxes the positivity condition, by replacing the lower bound by a relative lower bound. Second, we also discuss Kucerovsky's theorem in the context of half-closed modules, which generalise unbounded Kasparov modules to symmetric (rather than self-adjoint) operators. In order to deal with the positivity condition for such non-self-adjoint operators, we introduce a fairly general localisation procedure, which (using a suitable approximate unit) provides a 'localised representative' for the KK-class of a half-closed module. Using this localisation procedure, we then prove several variants of Kucerovsky's theorem for half-closed modules. A distinct advantage of the localised approach, also in the special case of self-adjoint operators (i.e., for unbounded Kasparov modules), is that the (global) positivity condition in Kucerovsky's original theorem is replaced by a (less restrictive) 'local' positivity condition, which is closer in spirit to the well-known Connes-Skandalis theorem in the bounded picture of KK-theory.
