Frobenius structures over Hilbert \(C^*\)-modules. (English) Zbl 1406.46039
The authors study the monoidal dagger category of Hilbert \(C^*\)-modules over a commutative \(C^*\)-algebra from the perspective of categorical quantum mechanics. A geometric description of Hilbert modules in terms of bundles is given in Section 4. In Section 5, the authors study dual objects and prove that dual objects are the finitely presented projective Hilbert \(C^*\)-modules. Dagger Frobenius structures are investigated in Section 6. Nontrivial examples of both commutative and central dagger Frobenius structures are also given. In Section 7, the authors apply bundle perspectives to dagger Frobenius structures and show that dagger Frobenius structures correspond to finite-dimensional \(C^*\)-algebras that vary continuously over the base space. Commutative special dagger Frobenius structures in the category of Hilbert modules are completely characterized in Section 8. The authors, in Section 9, reduce the study of special dagger Frobenius structures to the study of central ones and commutative ones by proving a transitivity theorem. Section 10 is devoted to the study of kernels. It turns out that the existence of kernels is related to clopen subsets and disconnectedness properties of the base space.
Reviewer: Wu Jing (Fayetteville)
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