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

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.

Keywords:

Superselection Theory, QFT on curved spacetimes