半群算子代数的Fell吸收原理
Fell's absorption principle for semigroup operator algebras
摘要：
Fell的吸收原理指出群的左正则表示吸收群的任何酉表示，在一个弱化形式下，这个结果可以推广到群的右LCM子幺半群的左正则表示和它的Nica协变等距表示，但是如果半群不满足独立性，这个结果就失效了。本文利用左正则表示的一个改进形式，将Fell的吸收原理推广到群$G$的任意子幺半群$P$.李半群$C^*$ -代数$C^*_s(P)$及其表示在本文中自然出现.使用增强的左正则表示，我们不仅提供了一个非常具体的介绍减少对象$C^*_s(P)$，但我们也获得了非常透明的证明早期的结果。我们还解决了非自伴理论，我们表明，非自伴对象连接到增强的左正则表示与左正则表示。我们得到了一个非自伴版本的Fell的吸收原理涉及的张量代数的半群，我们用它来改善最近的结果Cloudman和Dor-On的剩余有限维的某些$C^*$ -代数与这样的张量代数。我们还给出了另一个证明的存在性的$C^*$ -代数，这是共通用的等变，Li-协变表示的一个submonoid $P$的一个组$G$。
英文摘要：
Fell's absorption principle states that the left regular representation of a group absorbs any unitary representation of the group when tensored with it. In a weakened form, this result carries over to the left regular representation of a right LCM submonoid of a group and its Nica covariant isometric representations but it fails if the semigroup does not satisfy independence. In this paper we explain how to extend Fell's absorption principle to an arbitrary submonoid $P$ of a group $G$ by using an enhanced version of the left regular representation. Li's semigroup $C^*$-algebra $C^*_s(P)$ and its representations appear naturally in our context. Using the enhanced left regular representation, we not only provide a very concrete presentation for the reduced object for $C^*_s(P)$ but we also obtain very transparent proofs of earlier results. We also address the non-selfadjoint theory and we show that the non-selfadjoint object attached to the enhanced left regular representation coincides with that of the left regular representation. We obtain a non-selfadjoint version of Fell's absorption principle involving the tensor algebra of a semigroup and we use it to improve recent results of Clouatre and Dor-On on the residual finite dimensionality of certain $C^*$-algebras associated with such tensor algebras. We also give yet another proof for the existence of a $C^*$-algebra which is co-universal for equivariant, Li-covariant representations of a submonoid $P$ of a group $G$.
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