拟格序群上的Fell丛与紧对齐乘积系的$\mathrm{C}^*$ -代数
Fell bundles over quasi-lattice ordered groups and $\mathrm{C}^*$-algebras of compactly aligned product systems
摘要：
我们定义了拟格序群上Fell丛的半饱和性和正交性的概念。证明了Hilbert双模的紧对齐乘积系统只要是可分解的，就可以自然地推广到半饱和正交Fell丛。反之，半饱和正交Fell丛完全由正纤维决定，其横截面$\mathrm{C}^*$ -代数同构于Hilbert双模可积系的相对Cuntz-Pimsner代数.我们表明，这种对应关系的一部分是等价的bicategories和使用这概括了几个结果的迈耶和作者的上下文中的单一对应关系。利用相对Cuntz-Pimsner代数的函子性研究了紧对齐乘积系上的$\mathrm{C}^*$ -代数与Morita等价$\mathrm{C}^*$ -代数之间的Morita等价性.
英文摘要：
We define notions of semi-saturatedness and orthogonality for a Fell bundle over a quasi-lattice ordered group. We show that a compactly aligned product system of Hilbert bimodules can be naturally extended to a semi-saturated and orthogonal Fell bundle whenever it is simplifiable. Conversely, a semi-saturated and orthogonal Fell bundle is completely determined by the positive fibres and its cross sectional $\mathrm{C}^*$-algebra is isomorphic to a relative Cuntz--Pimsner algebra of a simplifiable product system of Hilbert bimodules. We show that this correspondence is part of an equivalence between bicategories and use this to generalise several results of Meyer and the author in the context of single correspondences. We apply functoriality for relative Cuntz--Pimsner algebras to study Morita equivalence between $\mathrm{C}^*$-algebras attached to compactly aligned product systems over Morita equivalent $\mathrm{C}^*$-algebras.
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