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

TL;DR

This paper investigates the structure and representation theory of nets of C*-algebras over non-upward directed posets, introducing C*-net bundles and classifying nets via fundamental group actions, with implications for conformal nets.

Contribution

It introduces the concept of C*-net bundles, classifies nets using fundamental group actions, and establishes a link between injectivity and faithful representations in the context of non-directed posets.

Findings

Any net embeds into a unique enveloping C*-net bundle.

Injectivity of nets is equivalent to the existence of faithful representations.

Examples of nets are provided that exhaust the classification.

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.