发表机构
Jilin University; Hangzhou Normal University（吉林大学; 杭州师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明等价étale群胚的约化与全$L^p$-算子代数具有Morita等价性，并比较链接代数，应用包括$L^p$版Green定理与$K$-理论不变性。
AI 中文摘要
我们证明了具有仿紧单位空间的等价局部紧、局部Hausdorff、étale群胚，对于每个$p\in[1,\infty]$，其约化与全$L^p$-算子代数都是Morita等价的。约化定理需要配对值近似恒等元，反映了约化范数在$1<p<\infty$时无条件性的失效；而全定理则依赖于通过膨胀空间表示证明的全-闭约化定理。对于$1\leq p<\infty$，我们还比较了约化Morita等价的具体$p$-链接代数与链接群胚的约化$L^p$-算子代数：典型比较映射是压缩的、单射的，且一般具有稠密值域，在$p=2$时是等距同构。我们进一步研究了由适当étale群胚对应产生的Morita循环，包括在适当扩展假设下的约化情形。应用包括Green对称印记定理的$L^p$版本、粗群胚和逆半群的结果，以及相应$K$-理论的不变性。
英文摘要
We prove that equivalent locally compact, locally Hausdorff, étale groupoids with paracompact unit spaces have Morita equivalent reduced and full $L^p$-operator algebras for every $p\in[1,\infty]$. The reduced theorem requires pairing-valued approximate identities, reflecting the failure of unconditionality of the reduced norm for $1<p<\infty$, while the full theorem rests on a full-clopen reduction theorem proved by dilating spatial representations. For $1\leq p<\infty$, we also compare the concrete $p$-linking algebra of the reduced Morita equivalence with the reduced $L^p$-operator algebra of the linking groupoid: the canonical comparison is contractive, injective, and has dense range in general, and is an isometric isomorphism for $p=2$. We further study Morita cycles arising from proper étale groupoid correspondences, including the reduced case under suitable extension hypotheses. Applications include an $L^p$ version of Green's symmetric imprimitivity theorem, results for coarse groupoids and inverse semigroups, and invariance of the corresponding $K$-theory.
Comments75 pages
