拟离散局部紧量子群
Quasi-Discrete Locally Compact Quantum Groups
摘要：
考虑一个C* 代数$A$具有一个余乘法$Δ$ .这个对通常被认为是局部紧量子半群。当这些笔记写于1993年时，根本不清楚这对是一个“局部紧量子群”的余乘的额外假设应该是什么。直到1999年，由于Kustermans和Vaes的工作，这一点才变得清晰起来。在紧凑的情况下，然而，而自然的条件下制定的Woronowicz和一个很好的概念，一个紧凑的量子群是在1993年。在这些注记中，我们考虑另一类局部紧量子群。我们假设$A$中存在一个非零元素$h$，对$A$中的所有$a$满足$Δ(a)(1\otimes h)=a \otimes h$。我们加上一些自然条件，我们说的“准离散局部紧量子群”。我们还讨论了离散的情况。证明了Haar测度、正则表示、基本酉的存在性，并得到了约化对偶.准离散情况原则上比离散情况更一般。在这些笔记写好后不久，Kustermans表明，事实上，这两个类是重合的。由于这个原因，这些说明没有发表。尽管如此，一些结果和技术似乎是有用的，在最近的工作中，我们遇到了类似的设置。因此，我们决定在网上提供这些说明。我们在导言的末尾添加了一些评论，并更新了参考列表。但除了这些小的改动之外，这些说明仍然是1993年编写和分发的。
英文摘要：
Consider a C*-algebra $A$ with a comultiplication $Δ$. This pair is usually thought of as locally compact quantum semi-group. When these notes were written, in 1993, it was not at all clear what the extra assumptions on the comultiplication should be for this pair to be a 'locally compact quantum group'. This only became clear in 1999 thanks to the work of Kustermans and Vaes. In the compact case however, rather natural conditions are formulated by Woronowicz and a good notion of a compact quantum group was available in 1993. In these notes, we consider another class of locally compact quantum groups. We assume the existence of a non-zero element $h$ in $A$ satisfying $Δ(a)(1\otimes h)=a \otimes h$ for all $a$ in $A$. We add some natural conditions and we speak of a 'quasi-discrete locally compact quantum group'. We also discuss the discrete case. We prove the existence of the Haar measure, the regular representation, the fundamental unitary and we obtain the reduced dual. The quasi-discrete case is, in principle, more general than the discrete case. Shortly after these notes were written, Kustermans showed that in fact, the two classes coincided. For this reason, these notes have not been published. Nevertheless, some of the results and techniques seem to be usful and in recent work, we came across similar settings. Therefore, we have decided to make these notes available on the net. We have added some comments at the end of the introduction and we have updated the reference list. But apart from these minor changes, the notes are still as they were written and distributed in 1993.
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