$W^*$-范畴是von Neumann
$W^*$-categories are von Neumann
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中文总结 AI 辅助
本文针对$W^*$-范畴表示论中的文献空白，利用对偶TRO理论修复，并提出允许非等距预对偶且要求双模性质的替代公理化，涉及自对偶希尔伯特$C^*$-模。
中文摘要 AI 辅助
一个$C^*$-范畴之于一个$C^*$-代数，正如一个群胚之于一个群。一个例子是希尔伯特空间与有界线性映射构成的范畴，事实上，每个$C^*$-范畴都可以表示为该范畴的一个子$*$-范畴。类似地，一个$W^*$-范畴是一个$C^*$-范畴，其中每个对象都有预对偶，我们可能期望在希尔伯特空间上有一个表示论，使得所得的态射空间是$\sigma$-弱闭的。我们在文献中发现了这里的一个空白，源于一个（也许令人惊讶的）事实：一个$C^*$-代数可以有一个非等距的预对偶，但仍然不是$W^*$-代数。利用对偶TRO（三元算子环）的理论，我们修复了这一空白，并在此过程中，也为文献中关于对偶TRO的论证提供了进一步的理由。受对偶巴拿赫代数理论的启发，我们提出了$W^*$-范畴的一种替代公理化，其中我们允许同态空间的非等距预对偶，但要求它们额外在某种意义上是双模。这涉及自对偶希尔伯特$C^*$-模的理论。
英文摘要
A $C^*$-category is to a $C^*$-algebra what a groupoid is to a group. An example is the category of Hilbert spaces and bounded linear maps, and indeed, every $C^*$-category can be represented as a sub-$*$-category of this. Analogously, a $W^*$-category is a $C^*$-category where every object has a predual, and we might expect a representation theory on Hilbert spaces where the resulting morphism spaces are $σ$-weakly closed. We identify a gap in the literature here, stemming from the (perhaps surprising) fact that a $C^*$-algebra can have a non-isometric predual and yet not be a $W^*$-algebra. Using the theory of dual TROs (ternary rings of operators) we repair this gap, and along the way, also give further justification to arguments in the literature about dual TROs. Motivated by the theory of Dual Banach Algebras, we offer an alternative axiomatisation of $W^*$-categories where we allow non-isometric preduals of the hom spaces, but require that they are additionally bimodules in a certain sense. This involves the theory of self-dual Hilbert $C^*$-modules.
