C∗-algebras and their Representations

E8Petrie

Definition. A C∗-algebra is a pair consisting of a ∗-algebra U and a norm

∥ · ∥ : A → C such that
∥AB∥ ≤ ∥A∥ · ∥B∥, ∥A∗A∥ = ∥A∥2,

∀ A, B ∈ A. We usually use A to denote the algebra and its norm.

Definition. A state ω on A is a linear functional such that ω(A∗A) ≥ 0 ∀ A ∈ U, and ω(I) = 1.

Definition. A state ω of U is said to be mixed if ω = 1/2(ω12) with ω1 ≠ ω2. Otherwise ω is said to be pure.

Definition. Let U be a C∗-algebra. A representation of U is a pair (H,π), where H is a Hilbert space and π is a ∗-homomorphism of U into B(H). A representation (H,π) is said to be irreducible if π(U) is weakly dense in B(H). A representation (H,π) is said to be faithful if π is an isomorphism.

Definition. Let (H, π) and (K, φ) be representations of a C∗-algebra U. Then (H,π) and (K,φ) are said to be:

  1. unitarily equivalent if there is a unitary U : H → K such that Uπ(A) = φ(A)U for all A ∈ U.
  2. quasiequivalent if the von Neumann algebras π(U)′′ and φ(U)′′ are ∗-isomorphic.
  3. disjoint if they are not quasiequivalent.

Definition. A representation (K, φ) is said to be a subrepresentation of (H, π) just in case there is an isometry V : K → H such that π(A)V =Vφ(A) ∀ A ∈ U.

Two representations are quasiequivalent iff they have unitarily equivalent subrepresentations.

The Gelfand-Naimark-Segal (GNS) theorem shows that every C∗-algebraic state can be represented by a vector in a Hilbert space.

Theorem:

(GNS). Let ω be a state of U. Then there is a representation (H,π) of U, and a unit vector Ω ∈ H such that:

1. ω(A)=⟨Ω, π(A)Ω⟩, ∀ A ∈ U;

2. π(U)Ω is dense in H.

Furthermore, the representation (H,π) is the unique one (up to unitarily equivalence) satisfying the two conditions.

Proof:

We construct the Hilbert space H from equivalence classes of elements in U, and the representation π is given by the action of left multiplication. In particular, define a bounded sesquilinear form on U by setting

⟨A, B⟩ω = ω(A∗B), A, B ∈ A.

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