16 research outputs found
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
The Haagerup Property for Discrete Measured Groupoids
International audienceWe define the Haagerup property in the general context of countable groupoids equipped with a quasi-invariant measure. One of our objectives is to complete an article of Jolissaint devoted to the study of this property for probability measure preserving countable equivalence relations. Our second goal, concerning the general situation, is to provide a definition of this property in purely geometric terms, whereas this notion had been introduced by Ueda in terms of the associated inclusion of von Neumann algebras. Our equivalent definition makes obvious the fact that treeability implies the Haagerup property for such groupoids and that it is not compatible with Kazhdan property (T)