Abstract. A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of the field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.Mathematics Subject Classification. 81V10, 81T05, 14F40

We construct the crossed product A ⋊ E ρ N of a C(X) -algebra A by an endomorphism ρ , in such a way that ρ becomes induced by the bimodule E of continuous sections of a vector bundle E → X . Some motivating examples for such a construction are given. Furthermore, we study the C*-algebra of G -invariant elements of the Cuntz-Pimsner algebra OE associated with E , where G is a (noncompact, in general) group acting on E . In particular, the C*-algebra of invariant elements w.r.t. the action of the group of special unitaries of E is a crossed product in the above sense. We also study the analogous construction on certain Hilbert bimodules, called 'noncommutative pullbacks'.

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 has a canonical morphism into a 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 such that the canonical morphism is faithful. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity to a generalized ech cocycle of the net, and this allows us to give examples of nets exhausting the above classification. Using these results we have shown, in another paper, that any conformal net over S (1) is injective

In algebraic quantum field theory the spacetime manifold is replaced by a suitable base for its topology ordered under inclusion. We explain how certain topological invariants of the manifold can be computed in terms of the base poset. We develop a theory of connections and curvature for bundles over posets in search of a formulation of gauge theories in algebraic quantum field theory

The notion of extension of a given C*-category C by a C*-algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging to the initial category. It is shown that the Doplicher-Roberts algebra (DR-algebra in the following) associated to an object in the extension of a strict tensor C*-category is a continuous field of DR-algebras coming from the initial one. In the case of the category of the hermitian vector bundles over Ω the general result implies that the DR-algebra of a vector bundle is a continuous field of Cuntz algebras. Some applications to Pimsner C*-algebras are given.