A new light on nets of C*-algebras and their representations

Abstract

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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• Renormalized Quantum Yang-Mills Fields in Curved Spacetime
• Structure and Classification of Superconformal Nets
• Quantum charges and spacetime topology: The emergence of new superselection sectors
• A theory of bundles over posets
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Cited by in corpus (10)

• Representations of Conformal Nets, Universal C*-Algebras and K-Theory
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• Algebraic field theory operads and linear quantization
• Presheaves of superselection structures in curved spacetimes
• The C(X)-algebra of a net and index theory
• The K-homology of nets of C*-algebras
• QED Representation for the Net of Causal Loops
• Homotopy theory of net representations
• Background potentials and superselection sectors
• Aharonov-Bohm superselection sectors