2022/01/26 by Alexandru Chirvăsitu, Chirvasitu, Alexandru
Computer Science · Mathematics · #20G42 #22D05 #22D25 #22D55 #46L67 #Advanced Algebra and Logic #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2201.10939
openalex publication_date 2022/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let ℍ\trianglelefteq\mathbbG be a closed normal subgroup of a locally compact quantum group. We introduce a strictly positive group-like element affiliated with L∞(\mathbbG) that, roughly, measures the failure of \mathbbG to act measure-preservingly on ℍ by conjugation. The triviality of that element is equivalent to the condition that \mathbbG and \mathbbG/ℍ have the same modular element, by analogy with the classical situation. This condition is automatic if ℍ≤ \mathbbG is central, and in general implies the unimodularity of ℍ. We also describe a bijection between strictly positive group-like elements δ affiliated with C0(\mathbbG) and quantum-group morphisms \mathbbG→ (ℝ,+), with the closed image of the morphism easily described in terms of the spectrum of δ. This then implies that property-(T) locally compact quantum groups admit no non-obvious strictly positive group-like elements.