Nica-Toeplitz algebras associated with right tensor C^*-precategories\n over right LCM semigroups

2016/11/25 by B. K. Kwaśniewski, Kwaśniewski, Bartosz K., Nadia S. Larsen +1
Computer Science · Mathematics · #46L05 (Secondary) #46L55 (Primary) #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1611.08525

openalex publication_date 2016/11/25 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We introduce and analyze the full \NT_\L(\K)\nand the reduced \NT_\L^r(\K) Nica-Toeplitz\nalgebra associated to an ideal \K in a right tensor\nC^*-precategory \L over a right LCM semigroup P. Our main\nresults are uniqueness theorems in the spirit of classical Coburn's theorem,\ngeneralizing uniqueness results for Toeplitz-type C^*-algebras associated to\nsingle C^*-correspondences, quasi-lattice ordered semigroups, and crossed\nproducts twisted by product systems of C^*-correspondences obtained by\nFowler, Laca and Raeburn.\n We formulate geometric conditions on a representation \Φ of \K\nso that the C^*-algebra it generates, C^*(\Φ(\K)), naturally\nlies between \NT_\L^r(\K) and\n\NT_\L(\K). Under suitable amenability\nhypotheses, C^*(\Φ(\K)) and\n\NT_\L(\K) are isomorphic. The geometric\nconditions are necessary for our uniqueness result when the right tensoring\npreserves \K and in general they capture uniqueness of the\nC^*-algebra generated by a natural extension of \Φ to \L. In\nparticular, the latter algebra could be viewed as a Doplicher-Roberts version\nof \NT_\L(\K).\n

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