# Fréchet Spaces and Presheave Morphisms.

A topological vector space V is both a topological space and a vector space such that the vector space operations are continuous. A topological vector space is locally convex if its topology admits a basis consisting of convex sets (a set A is convex if (1 – t) + ty ∈ A ∀ x, y ∈ A and t ∈ [0, 1].

We say that a locally convex topological vector space is a Fréchet space if its topology is induced by a translation-invariant metric d and the space is complete with respect to d, that is, all the Cauchy sequences are convergent.

A seminorm on a vector space V is a real-valued function p such that ∀ x, y ∈ V and scalars a we have:

(1) p(x + y) ≤ p(x) + p(y),

(2) p(ax) = |a|p(x),

(3) p(x) ≥ 0.

The difference between the norm and the seminorm comes from the last property: we do not ask that if x ≠ 0, then p(x) > 0, as we would do for a norm.

If {pi}{i∈N} is a countable family of seminorms on a topological vector space V, separating points, i.e. if x ≠ 0, there is an i with pi(x) ≠ 0, then ∃ a translation-invariant metric d inducing the topology, defined in terms of the {pi}:

d(x, y) = ∑i=1 1/2i pi(x – y)/(1 + pi(x – y))

The following characterizes Fréchet spaces, giving an effective method to construct them using seminorms.

A topological vector space V is a Fréchet space iff it satisfies the following three properties:

• it is complete as a topological vector space;
• it is a Hausdorff space;
• its topology is induced by a countable family of seminorms {pi}{i∈N}, i.e., U ⊂ V is open iff for every u ∈ U ∃ K ≥ 0 and ε > 0 such that {v|pk(u – v) < ε ∀ k ≤ K} ⊂ U.

We say that a sequence (xn) in V converges to x in the Fréchet space topology defined by a family of seminorms iff it converges to x with respect to each of the given seminorms. In other words, xn → x, iff pi(xn – x) → 0 for each i.

Two families of seminorms defined on the locally convex vector space V are said to be equivalent if they induce the same topology on V.

To construct a Fréchet space, one typically starts with a locally convex topological vector space V and defines a countable family of seminorms pk on V inducing its topology and such that:

1. if x ∈ V and pk(x) = 0 ∀ k ≥ 0, then x = 0 (separation property);
2. if (xn) is a sequence in V which is Cauchy with respect to each seminorm, then ∃ x ∈ V such that (xn) converges to x with respect to each seminorm (completeness property).

The topology induced by these seminorms turns V into a Fréchet space; property (1) ensures that it is Hausdorff, while the property (2) guarantees that it is complete. A translation-invariant complete metric inducing the topology on V can then be defined as above.

The most important example of Fréchet space, is the vector space C(U), the space of smooth functions on the open set U ⊆ Rn or more generally the vector space C(M), where M is a differentiable manifold.

For each open set U ⊆ Rn (or U ⊂ M), for each K ⊂ U compact and for each multi-index I , we define

||ƒ||K,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, ƒ ∈ C(U)

Each ||.||K,I defines a seminorm. The family of seminorms obtained by considering all of the multi-indices I and the (countable number of) compact subsets K covering U satisfies the properties (1) and (1) detailed above, hence makes C(U) into a Fréchet space. The sets of the form

|ƒ ∈ C(U)| ||ƒ – g||K,I < ε

with fixed g ∈ C(U), K ⊆ U compact, and multi-index I are open sets and together with their finite intersections form a basis for the topology.

All these constructions and results can be generalized to smooth manifolds. Let M be a smooth manifold and let U be an open subset of M. If K is a compact subset of U and D is a differential operator over U, then

pK,D(ƒ) := supx∈K|D(ƒ)|

is a seminorm. The family of all the seminorms  pK,D with K and D varying among all compact subsets and differential operators respectively is a separating family of seminorms endowing CM(U) with the structure of a complete locally convex vector space. Moreover there exists an equivalent countable family of seminorms, hence CM(U) is a Fréchet space. Let indeed {Vj} be a countable open cover of U by open coordinate subsets, and let, for each j, {Kj,i} be a countable family of compact subsets of Vj such that ∪i Kj,i = Vj. We have the countable family of seminorms

pK,I := supx∈K |(∂|I|/∂xI (ƒ)) (x)|, K ∈  {Kj,i}

inducing the topology. CM(U) is also an algebra: the product of two smooth functions being a smooth function.

A Fréchet space V is said to be a Fréchet algebra if its topology can be defined by a countable family of submultiplicative seminorms, i.e., a countable family {qi)i∈N of seminorms satisfying

qi(ƒg) ≤qi (ƒ) qi(g) ∀ i ∈ N

Let F be a sheaf of real vector spaces over a manifold M. F is a Fréchet sheaf if:

(1)  for each open set U ⊆ M, F(U) is a Fréchet space;

(2)  for each open set U ⊆ M and for each open cover {Ui} of U, the topology of F(U) is the initial topology with respect to the restriction maps F(U) → F(Ui), that is, the coarsest topology making the restriction morphisms continuous.

As a consequence, we have the restriction map F(U) → F(V) (V ⊆ U) as continuous. A morphism of sheaves ψ: F → F’ is said to be continuous if the map F(U) → F'(U) is open for each open subset U ⊆ M.

# von Neumann Algebras

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.