Origen de la transformación hacia la Nueva Gestión Pública

Theorem 3.2.13 ensures that the resolvent of the disordered Hamiltonian is contained in our observable algebra. If this were our only requirement for the observable alge- bra, then we could have simply taken the C∗-algebra generated by the resolvent of the (disordered) Hamiltonian. Because the quantum Hall effect involves a disordered Hamiltonian, current operators and the geometry of the momentum space, we require the larger crossed-product algebra. The algebra C(Ω)oθRd is also required to deter- mine the topological properties of higher-dimensional systems.

One of the strengths of Bellissard’s noncommutative Brillouin zone is that there is enough structure on A and the dense subalgebra A ∼=Cc(Ω×Rd) to define a calculus of sorts. This extra structure is of interest to us as we would like to consider the current operators Jk = i[H, Xk], where Xk is the position operator on L2(Rd) for k∈ {1, . . . , d}. In the quantum Hall example, such operators give the Hall current and come from a ‘noncommutative derivative’ of the Hamiltonian. The noncommutative calculus of A ⊂ A allows us to make sense of these derivatives. Furthermore, by

constructing an ‘integration theory’ on the algebra of observables, we can also consider the measurements of such current operators.

We start by defining a measure-theory on our algebra, which we do via a trace. We will consider two traces: an abstract trace defined on the algebraAand another coming from measurements in translation invariant systems. Under suitable hypotheses, we will show that these traces coincide.

Definition 3.2.15. Suppose the dynamical system (Ω,Rd, T) has an invariant Borel probability measure P. For f ∈ Aand f ≥0, we define

T(f) = Z

Ω

f(ω,0) dP(ω).

Lemma 3.2.16. The functionalT is a semifinite norm lower-semicontinous trace with

A ⊂Dom(T). If the support of P isΩ, then the trace T is faithful. Proof. We first check that

T(f∗f) = Z Ω (f∗f)(ω,0) dP(ω) = Z Ω Z Rd

eiθ(0∧y)f∗(ω, y)f(T−yω,0−y) dydP(ω) =

Z

Ω Z

Rd

f(T−yω,−y)f(T−yω,−y) dydP(ω) = Z Ω Z Rd |f(Tyω, y)|2dydP(ω),

which is finite and non-negative for f ∈ Cc(Ω×_Rd_{). Hence T is well-defined for any} positive f ∈ A.

It is a simple check thatT satisfies the linearity properties required for a trace. We then compute T(f f∗) = Z Ω Z Rd f(ω, y)f∗(T−yω,−y) dydP(ω) = Z Ω Z Rd f(ω, y)f(TyT−yω, y) dydP(ω) = Z Ω Z Rd |f(ω, y)|2dydP(ω), (3.7)

which is the same as T(f∗f) as P is invariant under the action of T. Therefore the functional T :A₊ →[0,∞] satisfies the conditions required to be a trace, where A₊ is the positive cone of A.

The trace is semifinite as it is well-defined on A, which is norm-dense in A. Next, suppose gn → g in norm. As kgk = supω∈Ωkπω(g)kB[L2_(Rd_)], we see that if g_n → g in norm, then by the definition of πω(gn), gn(ω, x) will converge pointwise to g(ω, x) almost everywhere. As gn, g ≥ 0, we can suppose gn = fn∗fn and g =f∗f. We then

compute T(f∗f) =Tlim n→∞f ∗ nfn = Z Ω Z Rd _n→∞lim fn(Tyω, y) 2 dydP(ω) ≤lim inf n→∞ Z Ω Z Rd |fn(Tyω, y)|2dydP(ω) = lim inf_n→∞ T(fn∗fn),

where we have used Fatou’s Lemma on the product measure defined by the Lebesgue measure on Rdand Pon Ω.

Finally, if supp(P) = Ω and T(f∗f) = T(f f∗) = 0, then Equation (3.7) implies that f(ω, x) = 0 for all ω and x (asf∗f is continuous).

Definition 3.2.17. For Λ⊂Rd open and convex, define TrΛ(T) = Tr(QΛT QΛ) where QΛ : L2(Rd) → L2(Λ) is the projection. Then taking an increasing sequence Λj with S

jΛj =Rd, thetrace per unit area onB[L2(Rd)] is given by Trar(T) = lim

j→∞ 1

|Λj|

TrΛj(T), T ≥0, where |Λj|denotes the Lebesgue measure of Λj.

Proposition 3.2.18. Let f ∈ A+. If P is an ergodic measure (that is, the only

functions inL2(Ω,P)such thatv(Txω) =v(ω) are constant functions), then for almost

all ω∈Ω,

T(f) = Trar[πω(f)].

Proof. Giveng∈ A, we know that [πω(g)ψ](x) =

Z

Rd

e−iθ(x∧y)g(T−xω, y−x)ψ(y) dy

soπω(g) is an integral operator with kernelkω(x, y) =e−iθ(x∧y)g(T−xω, y−x). Because Λ is bounded and kω(x, y) is continuous, πω(g) is Hilbert-Schmidt onL2(Λ) by [RS72, Theorem VI.23] for anyg∈Cc(Ω×Rd). Therefore we can say that the productπω(g∗g) is trace-class by [RS72, Theorem VI.22, part (h)] for g ∈ Cc(Ω×Rd). We can take the trace TrΛ by integrating along the diagonal [Sim05, Theorem 3.9]. Computing the trace forf =g∗g, TrΛ[πω(f)] = Z Λ kω(x, x) dx= Z Λ e−iθ(x∧x)f(T−xω, x−x) dx = Z Λ f(T−xω,0) dx.

As the action of Rd by T on Ω is P-measure preserving, a continuous version of Birkhoff’s Ergodic Theorem in higher dimensions [NZ79, Section 4] gives that

Trar[πω(f)] = lim j→∞ 1 |Λj|TrΛ[πω(f)] = limj→∞ 1 |Λj| Z Λj f(T−xω,0) dx = Z Ω f(ω,0) dP(ω) =T(f) for almost all ω.

Remark 3.2.19. We shall assume from now on that the probability measurePis ergodic under the action of Rd on Ω with supp(P) = Ω. In an abuse of notation, we will also denote the trace per unit volume byT, where T(f) =T(πω(f)) almost surely.

Definition 3.2.20. Forp≥1, denote by Lp(A,T) the completion ofA in the norm

kfkp = [T(|f|p)]1/p.

In particular, L2(A,T) is a Hilbert space with inner product hf1, f2i = T(f₁∗f2). The space L2(A,T) comes with a canonical representation πGN S : A → B[L2(A,T)] given by left multiplication.

Now that we have a measure theory on our algebra, we construct a differential structure. In the noncommutative framework, derivations on an algebra take the place of derivatives of functions.

Lemma 3.2.21. For all j ∈ {1, . . . , d}, the mapping (∂jf)(ω, x) = ixjf(ω, x) is a

∗-derivation on Cc(Ω×Rd).

Proof. The only claims that are not clear are the Leibniz rule∂j(f g) =∂j(f)g+f ∂j(g) and that∂j(f∗) = [∂j(f)]∗. By direct calculation,

[∂j(f)g) +f ∂j(g)](ω, x) = Z Rd eiθ(x∧y)(∂jf)(ω, y)g(T−yω, x−y) dy + Z Rd eiθ(x∧y)f(ω, y)(∂jg)(T−yω, x−y) dy = Z Rd

eiθ(x∧y)iyjf(ω, y)g(T−yω, x−y) dy +

Z

Rd

eiθ(x∧y)f(ω, y)i(xj−yj)g(T−yω, x−y) dy =

Z

Rd

eiθ(x∧y)i(yj+xj−yj)f(ω, y)g(T−yω, x−y) dy =ixj Z Rd eiθ(x∧y)f(ω, y)g(T−yω, x−y) dy =ixj(f g)(ω, x) = [∂j(f g)](ω, x). Also [∂j(f∗)](ω, x) =ixjf(T−xω,−x) =i(−xj)f(T−xω,−x) = (ixjf)∗(ω, x) = [∂j(f)]∗(ω, x) as required.

Because the derivations {∂j}d

j=1 commute onCc(Ω×Rd), we may exponentiate to obtain a d-parameter group of ∗-automorphsims on A given by

[ρk(f)](ω, x) =eik·xf(ω, x)

Lemma 3.2.22. Let X= (X1, . . . , Xd) be the position operator onL2(Rd) andf ∈ A.

Then for all k∈_Rd _{and j∈ {1, . . . , d},}

πω[ρk(f)] =e−ik·Xπω(f)eik·X, πω(∂jf) =−i[Xj, πω(f)].

Proof. We check for anyψ∈L2(Rd), [πω(ρk(f))ψ](x) = Z Rd e−iθ(x∧y)(ρkf)(T−xω, y−x)ψ(y) dy = Z Rd

e−iθ(x∧y)eik·(y−x)f(T−xω, y−x)ψ(y) dy =e−ik·X

Z

Rd

e−iθ(x∧y)f(T−xω, y−x)eik·yψ(y) dy =

h

e−ik·Xπω(f)eik·Xψ i (x). We also find [πω(∂jf)ψ](x) = Z Rd e−iθ(x∧y)(∂jf)(T−xω, y−x)ψ(y) dy = Z Rd

e−iθ(x∧y)i(yj−xj)f(T−xω, y−x)ψ(y) dy =−i xj Z Rd e−iθ(x∧y)f(T−xω, y−x)ψ(y) dy − Z Rd

e−iθ(x∧y)f(T−xω, y−x)yjψ(y) dy

=−i(Xjπω(f)ψ−πω(f)Xjψ) (x) = (−i[Xj, πω(f)]ψ)(x)

Because of the resultπω(∂jf) =−i[Xj, πω(f)], we will also denote∂j(a) =−i[Xj, a] foraa bounded operator on L2(Rd) witha·Dom(Xj)⊂Dom(Xj) andj∈ {1, . . . , d}. An immediate consequence of Lemma3.2.22is that if the resolvent of a Hamiltonian πω(f) = (λ−H)−1 is in the domain Dom(∂j) (as is the case forH0=PjKj2), then

∂j[πω(f)] =i[(λ−H)−1, Xj] = (λ−H)−1Jj(λ−H)−1.

Hence the differential structure on the algebra of observables allows us to detect infor- mation about the current operators.