In AQFT the observables are given by a causal net of algebras, usually -algebras. A concrete physical system corresponds to a state of the algebra of all observables, which leads, via the GNS construction, to a representation of this algebra on a concrete Hilbert space. In this way the familiar picture of quantum mechanics reappears. The interpretation of states and their representation as modelling concrete physical systems means that a systematic study of all representation of a given algebra of observables is central to AQFT.
A representation is faithful if its kernel is trivial.
Given two representations on and on , if there is a unitary operator such that then the representations are unitarily equivalent. A linear map (not necessarily unitary) having this property is called an intertwiner or an intertwining map. If there is no nontrivial intertwiner the two representations and are called disjoint (or totally / completley different).
In the context of AQFT the term ‘intertwiner’ is mostly used in the specific sense defined here.
From the physical viewpoint unitarily equivalent representations describe the same system, so that the classification of not unitarily equivalent representations is an important topic.
If there is a subspace of the Hilbert space which is invariant under , that is , then the restriction of the representation to is again a representation of , it is called a subrepresentation of .
Given a family of representations, we can form the direct sum of the Hilbert spaces and define a new representation via . This is the direct sum of representations.
See at operator algebras.