Unitary representation

Let H be complex Hilbert space. The group of unitary operators on H is denoted by U(H). Let G be a locally compact group. A strongly continuous unitary representation π of G on H is a homomorphism (of groups) from G toU(H) which is strongly continuous, meaning that the mapsG→ H,

a7→π(a)v,

are continuous for all v ∈ H. The scalar product of H is denoted by h·,·i and is antilinear in the first and linear in the second argument (physics convention).

A strongly continuous unitary representation is not necessarily continuous in the norm topology. However, as the strong operator topology coincides with the weak operator topology on U(H), a unitary representation is strongly continuous if and only if it is weakly continuous. That is, for allv, w∈ H the mapsG→C,

a7→ hv, π(a)wi,

are continuous. All representations which occur in this thesis will be unitary and strongly continuous. Therefore, we will often omit the adjectives and simply write representations. If not otherwise stated, the representation space ofπ is denoted by H_π. To avoid confusions we will use the notation (π,H) when we want to refer to a certain Hilbert space H.

Each locally compact group admits four important examples of representations.

Definition 2.34. Let Gbe a locally compact group.

(i) The representation mapping every element of G to 1H, the identity of a Hilbert space H, is called the trivial representation.

(ii) The left regular representation λG of G on the Hilbert space L²(G) is given by

[λG(b)f](a) =f(b⁻¹a) ∀f ∈L²(G).

(iii) The right regular representationρ_G of G on the Hilbert spaceL²(G) is given

by

[ρG(b)f](a) = ∆(b)¹²f(ab) ∀f ∈L²(G).

(iv) Thetwo-sided regular representationτ_G of G×Gon the Hilbert spaceL²(G) is given by

[τG(b, c)f](a) = ∆(c)¹²f(b⁻¹ac) ∀f ∈L²(G).

Among the linear mappings between Hilbert spaces, those which are compatible with the group action play a special role in representation theory.

Definition 2.35. Let G be a locally compact group and let π,π₁, π₂ be unitary repre-sentations of G on Hilbert spacesH, H₁, H₂, respectively.

(i) A bounded operator T ∈ B(H₁,H₂) is called an intertwiner for π1 and π2 if it satisfies

T π₁(a) =π₂(a)T ∀a∈G.

(ii) The space of all intertwiners is denoted byC(π1, π2). It is a closed linear subspace of B(H₁,H₂).

(iii) If there exists an isometry T ∈ C(π₁, π₂) then π₁ is called a subrepresentation of π₂ and we writeπ₁≤π₂.

(iv) If there exists a unitary operatorU ∈ C(π₁, π₂)thenπ₁andπ₂ are calledunitarily equivalent or simply equivalent, and we write π1 ∼=π2.

(v) The space C(π) := C(π, π) is called the commutant of π. It consists of all operators T ∈ B(H) := B(H,H) which commute with the operators π(a) for all a∈G.

One of the main goals of representation theory is to analyze the structure of representa-tions. In particular, the focus is on the decomposition into atoms, i.e., into irreducible ones.

Definition 2.36. Let G be a locally compact group and π a unitary representation of G on a Hilbert space H. Let H₀ be closed subspace of H.

(i) H₀ is called invariant under π if π(a)H₀ ⊆ H₀ for all a∈G.

(ii) IfH₀6={0} andH₀ is an invariant subspace ofH then the restriction ofπ to H₀ is a subrepresentation of π and denoted by π|_H0.

(iii) π is called irreducible if the only invariant subspaces ofH are{0} and H. Other-wise, π is called reducible.

The following theorem is one of the most important results for checking if a given representation is irreducible or not.

Theorem 2.37 (Schur’s Lemma). Let Gbe a locally compact group.

(i) A unitary representation π of Gis irreducible if and only if C(π) =C·1.

(ii) Ifπ1andπ2 are irreducible unitary representations ofG, thenπ1andπ2 are equiv-alent if and only if C(π₁, π₂) is one-dimensional. If π₁ and π₂ are not equivalent then C(π₁, π₂) ={0}.

Definition 2.38. Let G be a locally compact group and π a unitary representation of G on a Hilbert space H. Denote the dual space of H by H. The contragredient representation π of π on His given by

π(a)φ:=φπ(a⁻¹) =φπ(a)^∗ for all a∈G andφ∈ H.

π and π are not necessarily equivalent. By Schur’s Lemma,π is irreducible if and only ifπ is.

Conversely, instead of passing to subrepresentations, one can also construct bigger representations using direct sums or tensor products. If (H₁,k · k₁) and (H₂,k · k₂) are two Hilbert spaces, then H₁⊕ H₂ with the norm

kv₁+v₂k²_⊕:=kv₁k²₁+kv₂k²₂, ∀v₁ ∈ H₁, v₂ ∈ H₂,

is a Hilbert space. The tensor product H₁⊗ H₂ is the space of all bounded operators T ∈ B(H₂,H₁) with finite norm

kTk²_⊗:= Tr[T^∗T] = Tr[T T^∗]<∞. (2.10) Within this picture a decomposable tensorv1⊗v2, withv1 ∈ H₁, v2 ∈ H₂ corresponds to the map

(v1⊗v2)(f) =v1·f(v2), ∀f ∈ H₂.

Remark 2.39. Note that this definition of tensor products of Hilbert spaces differs from the one given in Folland [31, Chp. A.3]. Therein the tensor product of Hilbert spaces H₁, H₂ is the set of all antilinear bounded operatorsT:H₂ → H₁ satisfying eq.(2.10).

Since there exists an antilinear isomorphism between H₂ and H₂, both definitions are isomorphic.

Definition 2.40. Let Gbe a locally compact group and let π1, π2 be unitary represen-tations of G on Hilbert spacesH₁, H₂, respectively.

(i) The direct sum π=π₁⊕π₂ of π₁ and π₂ is the unitary representation ofG on H₁⊕ H₂ given by

π(a)(v₁+v₂) =π₁(a)v₁+π₂(a)v₂ for all a∈G andv1 ∈ H₁, v2 ∈ H₂.

(ii) The (inner) tensor product π=π₁⊗π₂ of π₁ and π₂ is the unitary represen-tation of G onH₁⊗ H₂ given by

π(a)(v1⊗v2) =π1(a)v1⊗π2(a)v2

for all a∈G andv₁ ∈ H₁, v₂ ∈ H₂.

(iii) If(π1,H₁) and(π2,H₂) are representations of two locally compact groupsG1 and G2, then the (outer) tensor product π =π1 ×π2 of π1 and π2 is the unitary representation of G₁×G₂ onH₁⊗ H₂ defined by

π(a1, a2)(v1⊗v2) =π1(a1)v1⊗π2(a2)v2

for all a₁∈G₁, a₂ ∈G₂ and v₁ ∈ H₁, v₂ ∈ H₂.

Hilbert spaces have the property that every closed subspace has an orthogonal comple-ment. As a consequence, any proper subrepresentation of a unitary representation has a complement, as well.

Proposition 2.41. Let G be a locally compact group and π a unitary representation of G on a Hilbert space H. Let H₀ be a nontrivial invariant subspace of H (meaning {0}(H₀ (H). Then the orthogonal complementH^⊥₀ is a nontrivial invariant subspace of H and

π ∼=π|_H0 ⊕π|_H⊥ 0. 2.2.2 Induced representations

The aim of the inducing procedure is to construct a unitary representation (π,H_π) of G from a given unitary representation (χ,H_χ) of a closed subgroup H ≤G. The idea goes back to Frobenius [32] who developed the concept to obtain irreducible unitary representations of finite groups from subgroups which are well studied. During the last century the idea was developed much further and became one of the most important techniques in representation theory and abstract harmonic analysis. In this section we follow Kaniuth and Taylor [44].

There are different ways to describe induced representations. In the following let Gbe a locally compact group and H be a closed subgroup ofG. Denote the quotient space

G/H by X and its quotient map G→X by p. Let δ=δ^G_H = ^∆_∆^G_H^|H:H →R>0 and let χ be a unitary representation ofH.

Variant 1. The first variant of induced representations, we will use, is the abstract version. The representation space is constructed as follows.

Let F(G, H, δ¹²χ) be the set of all functionsF:G→ H_χ satisfying

• F is continuous (with respect to the norm topology onH_χ),

• p(supp(F)) is compact inX,

• F(ah) =δ(h⁻¹)¹²χ(h⁻¹)F(a), for alla∈G and h∈H.

For allF1, F2∈ F(G, H, δ¹²χ) the function

a7→ hF₁(a), F₂(a)i_χ

is a rho-function and, therefore, there exists a Radon measure ν_F1,F2 on X satisfying Z

X

ϕ(x) dν_F1,F2(x) =^Z

G

β(a)ϕ(p(a))hF₁(a), F₂(a)i_χda

for all ϕ ∈ C_c(X) and some Bruhat section β:G → R≥0. The measure ν_F1,F2(x) is finite as

ν_F1,F2(X) =^Z

G

β(a)hF₁(a), F₂(a)i_χda

and a 7→ β(a)hF₁(a), F₂(a)i_χ is a compactly supported continuous function. It is straightforward to check that the bilinear form

hF₁, F₂i=ν_F1,F2(X) =^Z

G

β(a)hF₁(a), F₂(a)i_χda

defines an inner product on F(G, H, δ¹²χ), which is independent of the choice of the Bruhat section. By completing F(G, H, δ¹²χ) we obtain a Hilbert space, which we denote byL²(G, H, δ¹²χ). The spaceL²(G, H, δ¹²χ) consists of all functionsF:G→ H_χ satisfying

• F is measurable and ρF:a7→ kF(a)k²_χ is a rho-function,

• and the measure νρF on X corresponding to the rho-function ρF is finite.

The norm ofF is given by

kFk² =νρF(X) =^Z

G

β(a)kF(a)k²_χda for any Bruhat section β.

Gacts on L²(G, H, δ¹²χ) by

[π₁(a)F](b) =F(a⁻¹b) for all a∈Gand almost allb∈G. π1 is a unitary as

kπ₁(a)Fk² =^Z

G

β(b)kF(a⁻¹b)k²_χdb=^Z

G

β(ab)kF(b)k²_χdb=kFk². Therefore, π₁ is a unitary representation.

Variant 2. The second variant is more natural because it makes directly use of the measure decomposition of the Haar measure of G over the Haar measure of H and a quasi-invariant measure on X (see Prop. 2.22). However, an additional ingredient is necessary, namely a fixed rho-function. Moreover, it is not obvious that the resulting representation does not depend on the choice of this rho-function.

Let ρ be a continuous strictly positive rho-function for (G, H) and let µ_ρ be the cor-responding quasi-invariant measure on X. Let F_ρ(G, H, χ) be the set of all functions F:G→ H_χ satisfying

• F is continuous (with respect to the norm topology onH_χ),

• p(supp(F)) is compact in X,

• F(xh) =χ(h⁻¹)F(x), for allx∈G and h∈H. For allF1, F2∈ F_ρ(G, H, χ) the function

a7→ hF₁(a), F₂(a)i_χ

is constant on cosets aH ⊆ G and, therefore, there exists a compactly supported continuous functionϕF1,F2 ∈Cc(X) such that

ϕ_F1,F2(p(a)) =hF₁(a), F₂(a)i_χ ∀a∈G.

The inner product on F_ρ(G, H, χ) is defined by hF₁, F2i_ρ=^Z

X

ϕF1,F2(x) dνρ(x)

and depends on the choice of the rho-function. Let L²_ρ(G, H, χ) be the Hilbert space obtained by completing F_ρ(G, H, χ). Gacts on L²_ρ(G, H, χ) by

[πρ(a)F(b)] = ρ(a⁻¹b) ρ(b)

!¹₂

F(a⁻¹b)

which is unitary as kπ_ρ(a)Fk²_ρ=^Z

Xσ(a⁻¹, x)ϕF,F(a⁻¹x) dνρ(x) =^Z

X

ϕF,F(x) dνρ(x) =kFk²_ρ, where we used that

dν_ρ(a⁻¹bH) =σ(a⁻¹, bH) dν_ρ(bH) = ρ(a⁻¹b)

ρ(b) dν_ρ(bH) (cf. Thm. 2.23).

Although, the Hilbert spaces L²_ρ(G, H, χ) and the representations π_ρ depend on the choice of the rho-functionρ, the representations (π_ρ, L²_ρ(G, H, χ)) are unitarily equiva-lent and, in particular, equivaequiva-lent toπ1 constructed inVariant 1. The unitary equiva-lenceL²(G, H, δ¹²χ)→L²_ρ(G, H, χ) is induced by the mapF(G, H, δ¹²χ)→ F_ρ(G, H, χ),

F 7→ρ¹²F for all F ∈ F(G, H, δ¹²χ).

Variant 3. Suppose that Gis second countable. By Cor. 2.32 there exists a regular Borel cross-section q: X → G of p and by Lem. 2.33 there exists a quasi-invariant measure ν_q such that

Z

G

f(a) da=^Z

X

Z

H

f(q(x)h)δ(h) dhdν_q(x) for all f ∈C_c(G).

Let L²_q(X, χ) be the space of measurable functionsf:X→ H_χ such that kfk²_q :=^Z

X

kf(x)k²_χdνq(x)<∞.

G acts onL²_q(X, χ) by

[π_q(a)f](x) =δ(q(x)⁻¹aq(a⁻¹x))¹²χ(q(x)⁻¹aq(a⁻¹x))f(a⁻¹x) which is unitary as

kπ_q(a)fk²_q=^Z

X

δ(q(x)⁻¹aq(a⁻¹x))kf(a⁻¹x)k²_χdν_q(x)

=^Z

X

ρ_q(a⁻¹q(x))

ρq(q(x)) kf(a⁻¹x)k²_χdν_q(x)

=^Z

X

kf(x)k²_χdν_q(x) =kfk²_q,

where we used thatρ_q(q(x)) = 1 for all x∈X and

δ(q(x)⁻¹aq(a⁻¹x)) =δ(q(x)⁻¹aq(a⁻¹x))ρq(q(a⁻¹x)) ρ_q(q(x))

| {z }

=1

= ∆H(q(a⁻¹x)⁻¹a⁻¹q(x))

∆G(q(a⁻¹x)⁻¹a⁻¹q(x))

ρq(q(a⁻¹x))

ρq(q(x)) = ρq(a⁻¹q(x)) ρq(q(x)) . The representationsπ_q depend on the choice ofqor, to be more precisely, onρ_q. Again, they are unitarily equivalent and equivalent to π₁ defined in Variant 1. The unitary equivalence is given by, L²(G, H, δ¹²χ)→L²_q(X, χ),

f 7→f ◦q.

Since all of these representations are equivalent we define the induced representation as follows.

Definition 2.42. Let H ≤ G be a closed subgroup of a locally compact group G and let χ be a unitary representation of H.

The representation π₁ constructed in Variant 1 is denoted by ind^GH(χ).

In application, it will depend on the situation which variant of the induced representa-tion will be used. Variant 1is very useful for technical applications and proofs dealing with groups which are not necessarily second countable. Variant 2andVariant 3are quite handy for explicit computations, in particular when dealing with second count-able groups. However, they depend on the choice of a quasi-invariant measure onX or on the choice of a Borel section X→G.

2.2.3 Properties of induced representations

The reason why the inducing procedure is so powerful is that in many cases it behaves as on would expect. Among others, it has the following properties, which will turn out to be very helpful. Proofs and details can be found in Kaniuth and Taylor [44].

The first property is that the contragredient of an induced representation is the repre-sentation induced from the contragredient one.

Lemma 2.43. Let χ be a representation of a closed subgroup H of G. Then ind^GH(χ)∼= ind^GH(χ).

The second property is that representations induced from direct sums are direct sums of representations induced from the individual ones.

Lemma 2.44. Let H be a closed subgroup of the locally compact group G, and let χ_i, i∈I, be any family of unitary representations of H. Then

ind^GH

M

i∈I

χ_i∼=^M

i∈I

ind^GH(χ_i).

In particular, if ind^GH(χ) is an irreducible representation ofG, thenχ is an irreducible one of H. The converse is not true.

The next property is calledinduction in stages. It states the following.

Theorem 2.45. Let H be a closed subgroup of G and let H0 be a closed subgroup of H. Let χ₀ be a representation of H₀. Then

ind^GH0(χ0)∼= ind^GH ind^HH0(χ0).

The last property states that the inducing procedure is compatible with taking outer tensor products of representations.

Theorem 2.46. Let G₁ andG₂ be locally compact groups and let H₁ ≤G₁ and H₂ ≤ G2 be closed subgroups. Let χ1 and χ2 be representations of H1 and H2, respectively.

Then

ind^G_H¹₁^×G×H²₂(χ₁×χ₂)∼= ind^GH¹1(χ₁)×ind^G_H²₂(χ₂).

2.3 The dual and the quasi-dual of a locally compact group

The aim of this section is to describe the decomposition of representations into (mul-tiples of) irreducible ones. Unfortunately, such a decomposition is not always unique and in those case it is not expedient to do that. However, we will see that it is always possible to get a decomposition into so-called primary representations, which are close to being (multiples of) irreducible ones. Indeed, this decomposition is unique and will turn out to be more useful.

The results presented here are taken from Folland [31, Chp. 7] and Dixmier [21, Chp. 5, Chp. 8]. The notation is mostly taken from Führ [33]. The main result of this section is the Plancherel Theorem. The version for unimodular groups and its proof can be found in Dixmier [21, Chp. 18]. The Plancherel Theorem for nonunimodular groups is taken from Tatsuuma [58] and Duflo and Moore [25].

2.3.1 Von Neumann algebras

There are several types of algebras which play an important role in representation theory. One of them are von Neumann algebras.

Definition 2.47. A *-algebra is a C-algebra endowed with a involution. That is a map a7→a^∗ satisfying

(a+b)^∗ =a^∗+b^∗, (λa)^∗=λa^∗, (ab)^∗=b^∗a^∗, (a^∗)^∗ =a, for all a, b∈A, λ∈C.

A von Neumann algebra is a *-subalgebra of the bounded linear operatorsB(H) on some Hilbert space H which contains 1H and is closed in the weak operator topology.

Von Neumann algebras often appear as commutants or bicommutants. Let H be a Hilbert space and let S⊆ B(H) be a subset. The commutantof S is the set

S⁰ ={T ∈ B(H)|ST =T S}.

Correspondingly, the bicommutant S⁰⁰ of S is the commutant of the commutant of S, i.e., S⁰⁰ = (S⁰)⁰. If S is closed under conjugation, i.e., S^∗ = S, then it is an easy exercise to prove thatS⁰ and S⁰⁰ are von Neumann algebras.

Commutants or bicommutants not only give easy examples of von Neumann algebras.

Via the famous von Neumann Density Theorem they provide a tool to verify that a given algebra is a von Neumann algebra.

Theorem 2.48 (Von Neumann Density Theorem). Let Hbe a Hilbert space and letA be a *-subalgebra of B(H) containing 1H.

Then, the following are equivalent.

• A is a von Neumann algebra.

• A is closed in the strong operator topology.

• A satisfies A=A⁰⁰.

The centerof a von Neumann algebraA is the commutative von Neumann algebra Z(A) =A∩A⁰ =A⁰∩A⁰⁰.

Commutative von Neumann algebras admit a spectral decomposition (cf. Dixmier [22, I.7.3 Thm. 1]).

Theorem 2.49. LetHbe a separable Hilbert space and letA⊆ B(H) be a commutative von Neumann algebra. Then there exists a standard Borel space M and a projection-valued measure P such that the map

f 7→

Z

M

f(m) dP(m) defines an isometric isomorphism from L^∞(M, µ) toA.

The connection between von Neumann algebras and representation theory can be illus-trated as follows.

Proposition 2.50. Let(π,H)be a representation of a second countable locally compact group Gon a separable Hilbert space H. Then Z(π) =C(π)∩ C(π)⁰=π(G)⁰∩π(G)⁰⁰ is a commutative von Neumann algebra.

Let P be the projection-valued measure on the standard Borel space M associated to Z(π) as in Thm. 2.49. For any measurable subset E ⊆M, PE =P(E) is an orthog-onal projection onto a closed subspace of H_E = P_E(H) ⊆ H. In particular, if H_E is nontrivial, then (π_E,H_E) defined by π_E(a) =P π(a)P^∗ is a subrepresentation ofπ. From Schur’s Lemma it follows that, for every irreducible representation (σ,H) of a second countable locally compact group G, the algebraZ(σ) has the form

Z(σ) =C·1H.

Therefore, M only consists of one point. In fact, this is what we would expect from Prop. 2.50 as σ has no subrepresentations. The converse is not true. For d ∈ N = N∪ {∞}letH_d be the separable Hilbert space of dimensiond. To be more precise, we say that

H_d=C^d ford∈N, (2.11)

H_∞=l²(N) =(an)n∈N∈C^N X

n∈N

|a_n|²<∞ . (2.12)

For any representation (π,H) let

d· H=H ⊗ H_d, d·π =π⊗1d

be the multiple ofH and π.

Proposition 2.51. Let π be a representation of G and d∈N. Then Z(π)∼=Z(d·π).

This is because

C(d·π) ={S⊗T |S ∈ C(π), T ∈ B(H)}⁰⁰, C(d·π)⁰ =S⊗1d|S∈ C(π)⁰ .

From Prop. 2.51 it follows that, ifσ is irreducible, thenZ(d·σ) =C·1d·H even though d·σ is not irreducible. So in some sense Z has information about subrepresentations of a representation π but it does not count multiplicity.

Definition 2.52. Let G be a locally compact group. A representation (π,H) of G is called a factor representation (or primary representation) if

π(G)⁰∩π(G)⁰⁰=C·1H.

Factor representations are those for which the measure space M consists only of one point.

Definition 2.53.

• Two representations π1 and π2 are disjoint if they have no nontrivial common subrepresentation.

• Two representationsπ1 and π2 arequasi-equivalent if there exists no subrepre-sentation ρ₁ ≤π₁ such that C(ρ₁, π₂) = 0, and no subrepresentationρ₂≤π₂ such thatC(ρ₂, π₁) = 0.

In that case we write π₁ ≈π₂.

Proposition 2.54. Two factor representations are either quasi-equivalent or disjoint.

One might expect or at least hope that a representation is a factor representation if and only if it is a multiple of an irreducible representation. Unfortunately, this is not always the case.

Definition 2.55. A factor representation which is quasi-equivalent to an irreducible representation is called a factor representation of type I. If every factor represen-tation is type I then G is called type I.

Example 2.56. Abelian locally compact groups and compact groups are type I. More-over, connected nilpotent, exponential, and semisimple Lie groups are type I. (See Führ [33, p. 72] for references.)

The Mautner group is a famous example of a solvable Lie group which is not type I.

(See Baggett [10] for details of its representation theory.)

The representation theory of groups of type I enjoys the pleasant features known from abelian locally compact groups and compact groups. For instance, every representation can be uniquely decomposed (in a generalized sense) into irreducible ones. For non-type I groups this is, in general, not true. In that case, to get similar results, one has to restrict attention to type I representations.

Definition 2.57. A representation π is called multiplicity-freeif C(π) is commuta-tive. π is called type I if it is quasi-equivalent to a multiplicity-free representation.

2.3.2 Direct integral of Hilbert spaces

Let M be a measurable space and let {H_m}_m∈M be a family of nonzero separable Hilbert spaces. Denote the scalar product and norm of H_m by h·,·i_m and k · k_m, respectively. The vector fields f ∈ ^Q_m∈MH_m are considered as functions mapping m∈M tof(m)∈ H_m.

Suppose there exists a countable set {v_j}_j∈N such that

• the functions m7→ hv_j(m), vk(m)i_a are measurable for all j, k,

• and {v_j(m)}_j∈N is total in H_m for all m∈M.

Then ({H_m}m∈M,{v_j}j∈N) is called ameasurable field of Hilbert spaces. A vector field f is called measurableifm 7→ hv_j(m), f(m)i_m is measurable for all j∈N. The set of measurable vector fields forms a vector space over the complex numbers.

Even if the family {v_j}_j∈N is chosen very complicated, there exists a family {e_j}_j∈N which has a particularly easy form and defines the same measurable structure.

Proposition 2.58. Let ({H_m}_m∈M,{v_j}_j∈N) be a measurable field of Hilbert spaces.

• There exist measurable vector fields {e_j}_j∈N such that {e_j(m)}j=1,...,dimH_m

is a complete orthonormal system and e_j(m) = 0 for j >dimH_m.

• {v_j}j∈N and {e_j}j∈N define the same measurable structure on {H_m}m∈M in the sense that a vector field f is measurable (with respect to {v_j}_j∈N) if and only if m7→ he_j(m), f(m)i_m is measurable for all j∈N.

• If f, g are measurable vector fields then m7→ hf(m), g(m)i_m is measurable.

This result basically follows from the fact that pointwise orthogonalization by the Gram-Schmidt process can be realized without affecting measurability. A detailed proof can be found in Folland [31, Prop. 7.19].

Given a measurable field of Hilbert spaces ({H_m}m∈M,{v_j}_j∈N) and aσ-finite measure µ onM the direct integral of the Hilbert spaces H_m,m∈M, denoted by

Z ⊕ M

H_mdµ(m), is the set of all measurable vector fieldsf satisfying

kfk² =^Z

M

kf(m)k²_mdµ(m)<∞, (2.13) where two measurable vector fields f, g are identified if kf−gk= 0.

Proposition 2.59. ^R_M^⊕H_mdµ(m) is a Hilbert space with respect to the scalar product hf, gi:=^Z

M

hf(m), g(m)i_mdµ(m), for all f, g∈^R_M^⊕H_mdµ(m).

For sure, the space^R_M^⊕H_mdµ(m) depends on the choice of{v_j}_j∈Nand one can easily find another family {w_j}_j∈N which defines a different measurable structure. For in-stance, by changing {v_j}_j∈N on a subset of M which is not measurable. However, it can be shown that there exists a unitary isomorphism between the two Hilbert spaces obtained from {v_j}_j∈N and {w_j}_j∈N which respects the direct integral structure. (For details see Folland [31, Chp. 7.4].) In that sense, it is not important which measurable structure is used to construct the direct integral of Hilbert spaces. Hence, they are often used without mentioning a measurable structure.

Next, we focus on operators on ^R_M^⊕H_mdµ(m). A field of operators {T(m)}_m∈M is called measurableif {T(m)f(m)}_m∈M is measurable for all measurable vector fields f. Measurability can also be characterized as follows.

Proposition 2.60. A field of operators {T(m)}_m∈M is measurable if and only ifm7→

hv_j(m), T(m)v_k(m)i_m is measurable for all j, k∈N.

Let µ be a σ-finite measure on M and let T = {T(m)}_m∈M be a measurable field of operators. T is called essentially bounded (i.e., up to null sets) if

kTk_∞:= ess sup

m∈M

kT(m)k_op <∞, wherek · k_op is the operator norm.

Proposition 2.61. LetT ={T(m)}_m∈M be an essentially bounded measurable field of operators. The operator ^R_M^⊕T(m) dµ(m) on ^R_M^⊕H_mdµ(m) defined by

Z ⊕ M

T(m) dµ(m)f

(m) :=T(m)f(m) is bounded and ^R_M^⊕T(m) dµ(m)

op =kTk_∞.

If T₁ ={T₁(m)}_m∈M and T₂={T₂(m)}_m∈M are essentially bounded measurable fields of operators such that ^R_M^⊕T1(m) dµ(m) = ^R_M^⊕T2(m) dµ(m) then T1(m) = T2(m) µ -almost everywhere.

The operator^R_M^⊕T(m) dµ(m) is called thedirect integral of the field of operatorsT. The space of all integral operators is denoted by

B^⊕(M, µ) =^{Z ⊕}

M

B(H_m) dµ(m)

and forms a von Neumann algebra contained in B^R_M^⊕H_mdµ(m). Its commutant B^⊕(M, µ)⁰ consists of all diagonal operators, i.e., the operators

Z ⊕ M

T(m) dµ(m)∈ B^⊕(M, µ)

of the form

T(m) =t(m)·1Hm, almost everywhere, for some essentially bounded measurable function t∈L^∞(M, µ).

For a Hilbert space H, let B₁(H) be the space of trace-class operators and let B₂(H) be the space of Hilbert-Schmidt operators on H. Denote by

B^⊕₁(M, µ)⊆ B^⊕(M, µ)

the Banach space of all direct integrals T =^R_M^⊕T(m) dµ(m)∈ B^⊕(M, µ) withT(m)∈ B₁(H_m) almost everywhere and

kTk₁ :=^Z

M

kT(m)k_m,1dµ(m)<∞, where k · k_m,1 denotes the trace-norm of B₁(H_m). Likewise, let

B^⊕₂(M, µ)⊆ B^⊕(M, µ)

be the direct integral of the Hilbert spaces{B₂(H_m)}_m∈M. That is the Hilbert space of all operatorsT =^R_M^⊕T(m) dµ(m)∈ B^⊕(M, µ) withT(m)∈ B₂(H_m) almost everywhere satisfying

kTk²₂ :=^Z

M

kT(m)k²_m,2dµ(m)<∞, where k · k_m,2 denotes the Hilbert-Schmidt-norm ofB₂(H_m).

LetGbe a second countable locally compact group and let{(π_m,H_m)}_m∈M be a family of representations. If for alla∈Gthe fields of operators{π_m(a)}_m∈M are measurable, then{(πm,H_m)}m∈M is called ameasurable field of representations. In that case, R⊕

Mπ_mdµ(m) given by Z ⊕

M

πmdµ(m)(a) =^{Z ⊕}

M

πm(a) dµ(m)

defines a unitary representation of Gon ^R_M^⊕H_mdµ(m). The representation Z ⊕

M

πmdµ(m)

is called thedirect integralof the representations {π_m}m∈M. 2.3.3 Decomposition of representations

The results of Chp. 2.3.1 and Chp. 2.3.2 can be used to decompose a representation as a direct integral.

Theorem 2.62. Let G be a second countable locally compact group, π a unitary rep-resentation of G on a separable Hilbert space H, and B a commutative von Neumann subalgebra of C(π).

There exists a standard Borel space M, a measurable field {H_m}_m∈M of Hilbert spaces, a measurable field {π_m}_m∈M of representations of G, and a unitary map U: H → R⊕

MH_mdµ(m), such that

• U π(a)U⁻¹ =^R_M^⊕π_m(a) dµ(m) for all a∈G

• and U BU⁻¹ is the algebra of diagonal operators on ^R_M^⊕H_mdµ(m).

• and U BU⁻¹ is the algebra of diagonal operators on ^R_M^⊕H_mdµ(m).

Im Dokument Continuous Wavelet Transformation on Homogeneous Spaces (Seite 33-0)