Frobenius structures over Hilbert C*-modules

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.Comment: 35 page

On the quantisation of points

In the study of quantales arising naturally in the context of -algebras, Gelfand quantales have emerged as providing the basic setting. In this paper, the problem of defining the concept of point of the spectrum of a -algebra A, which is one of the motivating examples of a Gelfand quantale, is considered. Intuitively, one feels that points should correspond to irreducible representations of A. Classically, the notions of topological and algebraic irreducibility of a representation are equivalent. In terms of quantales, the irreducible representations of a -algebra A are shown to be captured by the notion of an algebraically irreducible representation of the Gelfand quantale on an atomic orthocomplemented sup-lattice S, defined in terms of a homomorphism of Gelfand quantales to the Hilbert quantale of sup-preserving endomorphisms on S. This characterisation leads to a concept of point of an arbitrary Gelfand quantale Q as a map of Gelfand quantales into a Hilbert quantale , the inverse image homomorphism of which is an algebraically irreducible representation of Q on the atomic orthocomplemented sup-lattice S. The aptness of this definition of point is demonstrated by observing that in the case of locales it is exactly the classical notion of point, while the Hilbert quantale of an atomic orthocomplemented sup-lattice S has, up to equivalence, exactly one point. In this sense, the Hilbert quantale is considered to be a quantised version of the one-point space