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Register a new userThe K-homology of nets of C*-algebras
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Let X be a space, intended as a possibly curved spacetime, and A a precosheaf of C*-algebras on X. Motivated by algebraic quantum field theory, we study the Kasparov and Theta-summable K-homology of A interpreting them in terms of the holonomy equivariant K-homology of the associated C*-dynamical system. This yields a characteristic class for K-homology cycles of A with values in the odd cohomology of X, that we interpret as a generalized statistical dimension.
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We establish exact sequences in $KK$-theory for graded relative Cuntz-Pimsner algebras associated to nondegenerate $C^*$-correspondences. We use this to calculate the graded $K$-theory and $K$-homology of relative Cuntz-Krieger algebras of directed graphs for gradings induced by ${0,1}$-valued labellings of their edge sets.
We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^*$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For $C^*$-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph $C^*$-algebras. These include the Toeplitz algebra, Cuntz algebras, and $q$-deformed spheres.
Given a connected and locally compact Hausdorff space X with a good base K we assign, in a functorial way, a C(X)-algebra to any precosheaf of C*-algebras A defined over K. Afterwards we consider the representation theory and the Kasparov K-homology of A, and interpret them in terms, respectively, of the representation theory and the K-homology of the associated C(X)-algebra. When A is an observable net over the spacetime X in the sense of algebraic quantum field theory, this yields a geometric description of the recently discovered representations affected by the topology of X.
We consider the construction of twisted tensor products in the category of C*-algebras equipped with orthogonal filtrations and under certain assumptions on the form of the twist compute the corresponding quantum symmetry group, which turns out to be the generalised Drinfeld double of the quantum symmetry groups of the original filtrations. We show how these results apply to a wide class of crossed products of C*-algebras by actions of discrete groups. We also discuss an example where the hypothesis of our main theorem is not satisfied and the quantum symmetry group is not a generalised Drinfeld double.
The present paper deals with the question of representability of nets of C*-algebras whose underlying poset, indexing the net, is not upward directed. A particular class of nets, called C*-net bundles, is classified in terms of C*-dynamical systems having as group the fundamental group of the poset. Any net of C*-algebras embeds into a unique C*-net bundle, the enveloping net bundle, which generalizes the notion of universal C*-algebra given by Fredenhagen to nonsimply connected posets. This allows a classification of nets; in particular, we call injective those nets having a faithful embedding into the enveloping net bundle. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity to a generalized Cech cocycle of the net, and this allows us to give examples of nets exhausting the above classification. Using the results of this paper we shall show, in a forthcoming paper, that any conformal net over S^1 is injective.
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