nLab
representation of a C-star-algebra

Context

Representation theory

Operator algebra

Contents

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

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 π 1\pi_1 on H 1H_1 and π 2\pi_2 on H 2H_2, if there is a unitary operator U:H 1H 2U: H_1 \to H_2 such that Uπ 1=π 2UU \pi_1 = \pi_2 U then the representations are unitarily equivalent. A linear map UU (not necessarily unitary) having this property is called an intertwiner or an intertwining map. If there is no nontrivial intertwiner the two representations π 1\pi_1 and π 2\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 1H_1 of the Hilbert space HH which is invariant under π(A)\pi(A), that is π(A)(H 1)H 1\pi(A)(H_1) \subseteq H_1, then the restriction of the representation to H 1H_1 is again a representation of AA, it is called a subrepresentation of π\pi.

Definition

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

References

See at operator algebras.

Revised on April 3, 2013 23:24:52 by Urs Schreiber (82.169.65.155)