A Cuntz-Krieger Uniqueness Theorem for a Class of Left-Cancellative Categories
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We introduce a class of left-cancellative small categories for which there is a Cuntz-Krieger uniqueness theorem. The categories arise naturally as a generalization of directed graphs for which paths are allowed to have lengths which are ordinals. We will describe a collection of \(C^*\)-correspondences which generalize the usual correspondence for a directed graph. Under suitable conditions, these correspondences satisfy condition (S) of Eryüzlü and Tomforde, and we describe how a transfinite induction argument yields the uniqueness theorem.