$\mathbb{N}^k$上$C^*$ -对应乘积系统的完全相关表示
Finitely Correlated Representations of Product Systems of $C^*$-Correspondences over $\mathbb{N}^k$
摘要：
研究半群$\mathbb{N}^k$上对应的乘积系的等距表示，它们是有限维的极小扩张的完全余等距表示。我们证明了所有这样的表示存在一个唯一的最小循环共不变子空间。压缩的表示到这个子空间被证明是完整的酉不变。对于某一类图代数，证明了由这些表示生成的非自伴\textsc{wot} -闭代数包含到极小循环共不变子空间上的投影.这类代数包括自由半群代数。这一结果推广到一类高阶图代数，其中包括具有一个顶点的高阶图。
英文摘要：
We study isometric representations of product systems of correspondences over the semigroup $\mathbb{N}^k$ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be complete unitary invariant. For a certain class of graph algebras the nonself-adjoint \textsc{wot}-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex.
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