# 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.

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