von Neumann Algebras

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The standard definition of a von Neumann algebra involves reference to a topology, and it is then shown (by von Neumann’s double commutant theorem) that this topological condition coincides with an algebraic condition (condition 2 in the Definition 1.2). But for present purposes, it will suffice to take the algebraic condition as basic.

1.1 Definition. Let H be a Hilbert space. Let B(H) be the set of bounded linear operators on H in the sense that for each A ∈ B(H) there is a smallest nonnegative number ∥A∥ such that ⟨Ax, Ax⟩1/2 ≤ ∥A∥ for all unit vectors x ∈ H. [Subsequently we use ∥ · ∥ ambiguously for the norm on H and the norm on B(H).] We use juxtaposition AB to denote the composition of two elements A,B of B(H). For each A ∈ B(H) we let A∗ denote the unique element of B(H) such that ⟨A∗x, y⟩ = ⟨x,Ay⟩, for all x,y ∈ R.

1.2 Definition. Let R be a ∗-subalgebra of B(H), the bounded operators on the Hilbert space H. Then R is a von Neumann algebra if

1. I ∈ R,

2. (R′)′ = R,

where R′ = {B ∈ B(H): [B,A] =0, ∀ A ∈ R}

1.3 Definition. We will need four standard topologies on the set B(H) of bounded linear operators on H. Each of these topologies is defined in terms of a family of seminorms.

  • The uniform topology on B(H) is defined in terms of a single norm: ∥A∥ = sup{∥Av∥ : v ∈ H, ∥v∥ ≤ 1}, where the norm on the right is the given vector norm on H. Hence, an operator A is a limit point of the sequence (Ai)i∈N iff (∥Ai − A∥)i∈N converges to 0.
  • The weak topology on B(H) is defined in terms of the family {pu,v : u, v ∈ H} of seminorms where pu,v(A) = ⟨u,Av⟩. The resulting topology is not generally first countable, and so the closure of a subset S of B(H) is generally larger than the set of all limit points of sequences in S. Rather, the closure of S is the set of limit points of generalized sequences (nets) in S. A net (Ai)i∈I in B(H) converges weakly to A just in case (pu,v(Ai))i∈I converges to pu,v(A) ∀ u,v ∈ H.
  • The strong topology on B(H) is defined in terms of the family {pv : v ∈ H} of seminorms where pv(A) = ∥Av∥. Thus, a net (Ai)i∈I converges strongly to A iff (pv (Ai))i∈I converges to pv(A), ∀ v ∈ H.
  • The ultraweak topology on B(H) is defined in terms of the family {pρ : ρ ∈ T (H)} where T (H) is the set of positive, trace 1 operators (“density operators”) on H and pρ(A) = Tr(ρA).

Thus a net (Ai)i∈I converges ultraweakly to A just in case (Tr(ρAi))i∈I converges to Tr(ρA), ∀ ρ ∈ T (H).

If S is a bounded, convex subset of B(H), then the weak, ultraweak, and norm closures of S are the same.

For a ∗-algebra R on H that contains I, the following are equivalent:

(i) R is weakly closed;

(ii) R′′ = R. This is von Neumann’s double commutant theorem.

1.4 Definition. Let R be a subset of B(H). A vector x ∈ H is said to be cyclic for R just in case [Rx] = H, where Rx = {Ax : A ∈ R}, and [Rx] is the closed linear span of Rx. A vector x ∈ H is said to be separating for R just in case Ax = 0 and A ∈ R entails A = 0.

Let R be a von Neumann algebra on H, and let x ∈ H. Then x is cyclic for R iff x is separating for R′.

1.5 Definition. A normal state of a von Neumann algebra R is an ultraweakly continuous state letting R∗ denote the normal state space of R.

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