nLab representation of a C-star-algebra

Idea

This page collects a few notions and facts about representations of C-star-algebras with an eye towards their use in AQFT.

In AQFT the observables are given by a causal net of algebras, usually $C^*$-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.

Definition

Definition

A representation $\pi$ of a $C^*$-algebra $A$ is a star-representation on a Hilbert space $H$, hence a $*$-homomorphism from $A$ to the algebra of bounded operators on $H$.

Remark

The continuity of the representation is implied by the star-representation-property.

Definition

A representation is faithful if its kernel is trivial.

Definition

Given two representations $\pi_1$ on $H_1$ and $\pi_2$ on $H_2$, if there is a unitary operator $U: H_1 \to H_2$ such that $U \pi_1 = \pi_2 U$ then the representations are unitarily equivalent. A linear map $U$ (not necessarily unitary) having this property is called an intertwiner or an intertwining map. If there is no nontrivial intertwiner the two representations $\pi_1$ and $\pi_2$ 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.

Definition

If there is a subspace $H_1$ of the Hilbert space $H$ which is invariant under $\pi(A)$, that is $\pi(A)(H_1) \subseteq H_1$, then the restriction of the representation to $H_1$ is again a representation of $A$, it is called a subrepresentation of $\pi$.

Definition

Given a family $(\pi_i, H_i)$ of representations, we can form the direct sum $H := \oplus H_i$ of the Hilbert spaces and define a new representation $\pi := \oplus \pi$ via $\pi(A) | H_i := \pi_i$. This is the direct sum of representations.

Related concepts

• cyclic vector

• GNS construction

References

See at operator algebras.