2017/04/30 by Chris Heunen, Manuel L. Reyes · 11 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Commutative property #Commutative ring #Dagger #Duality (order theory) #Frobenius algebra #Frobenius theorem (differential topology) #Geometry #Mathematics #Noncommutative and Quantum Gravity Theories #Pure mathematics #math.CT #math.OA
paper · pdf · doi:10.1007/s00220-018-3166-0
published in Communications in Mathematical Physics 361(2), 787-824 (Springer Science+Business Media) · 35 pages
arxiv created 2018/04/10 · openalex publication_date 2018/06/09 · openalex created_date 2020/11/23 · arxiv updated 2020/12/03 · openalex updated_date 2026/08/06
We study the monoidal dagger category of Hilbert C*-modules over a commutative C*-algebra from the perspective of categorical quantum mechanics. The dual objects are the finitely presented projective Hilbert C*-modules. Special dagger Frobenius structures correspond to bundles of uniformly finite-dimensional C*-algebras. A monoid is dagger Frobenius over the base if and only if it is dagger Frobenius over its centre and the centre is dagger Frobenius over the base. We characterise the commutative dagger Frobenius structures as finite coverings, and give nontrivial examples of both commutative and central dagger Frobenius structures. Subobjects of the tensor unit correspond to clopen subsets of the Gelfand spectrum of the C*-algebra, and we discuss dagger kernels.