Reduced C^*-algebras of Product Systems -- an E₀-semigroup and a Groupoid perspective

2025/07/27 by Hossain, Md Amir, Sundar, S. · 1 citation
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2507.20319

Abstract

For Ore semigroups P with an order unit, we prove that there is a bijection between E₀-semigroups over P and product systems of C^*-correspondences over P^op. We exploit this bijection and show that the reduced C^*-algebra of a proper product system is Morita equivalent to the reduced crossed product of the associated semigroup dynamical system given by the corresponding E₀-semigroup. We appeal to the groupoid picture of the reduced crossed product of a semigroup dynamical system derived in [47] to prove that, under good conditions, the reduced C^*-algebra of a proper product system is nuclear/exact if and only if the coefficient algebra is nuclear/exact. We also discuss the invariance of K-theory under homotopy of product systems.

Citations

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Cited by

• Some remarks on Reduced C^*-algebras of semigroup dynamical systems and product systems

Related

• E₀-semigroups and product systems of W^*-bimodules
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• Product systems and their representations: an approach using Fock spaces and Fell bundles
• Groupoid Crossed Products of Continuous-Trace C*-Algebras