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

Giuseppe Ruzzi, Ezio Vasselli
May 18, 2010
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.
open access link Comm. Math. Phys. 312 (2012) 655-694
@article{Ruzzi:2010sj, author = "Ruzzi, Giuseppe and Vasselli, Ezio", title = "{A new light on nets of C*-algebras and their representations}", journal = "Commun. Math. Phys.", volume = "312", year = "2012", pages = "655-694", doi = "10.1007/s00220-012-1490-3", eprint = "1005.3178", archivePrefix = "arXiv", primaryClass = "math.OA", SLACcitation = "%%CITATION = ARXIV:1005.3178;%%" }

Keywords:
Superselection Theory, QFT on curved spacetimes