Pseudo-Differential Operators on GeneralType I Locally Compact Groups

Tesisentregada a la

Universidad de Chileen cumplimiento parcial de los requisitos

para optar al grado deMagíster en Ciencias Matemáticas

Facultad de Cienciaspor

Maximiliano Sandoval R.

Abril de 2017

Director de Tesis: Dr. Marius Mantoiu

FACULTAD DE CIENCIAS

UNIVERSIDAD DE CHILE

INFORME DE APROBACIÓN

TESIS DE MAGISTER

Se informa a la escuela de Postgrado de la Facultad de Ciencias que la Tesis de Magisterpresentada por el candidato

Maximiliano Andrés Sandoval RiveroHa sido aprobada por la comisión de Evaluación de la tesis como requisito para optar algrado de Magister en Ciencias Matemáticas, en el examen de Defensa Privada de Tesisrendido el día

Director de Tesis:Dr. Marius Mantoiu . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Comisión de Evaluación:

Dr. Radu Purice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Jorge Soto Andrade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dr. Eduardo Friedman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Mathematics is the artof giving the same name to different things.

HENRI POINCARÉ.

Agradecimientos

Agradezco enormemente a mis padres por su apoyo, que sin ellos no estaría aquí. Graciasa todos mis profesores durante la licenciatura.

Muchas gracias.

Este trabajo fue financiado por la beca CONICYT-PCHA/MagísterNacional/2016–22160383y fue parcialmente financiado por Núcleo Milenio de Física Matemática RC120002 y elproyecto Fondecyt 1120300.

iii

Resumen

In a recent paper by M. Măntoiu and M. Ruzhansky a global pseudo-differential calculus hasbeen developed for unimodular groups of type I. In the present thesis we generalize the mainresults to arbitrary locally compact groups of type I. Our methods involve defining suitableWeyl systems, Wigner transforms and the use of Plancherel’s theorem for non-unimodulargroups. We also give explicit constructions for the group of affine transformations of thereal line and Grélaud’s group.

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Contents

1 Framework 61.1 General remarks from functional analysis and measure theory . . . . . . . . 61.2 Direct integrals of Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Basic representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 The left and right regular representations . . . . . . . . . . . . . . . 121.4 Induced representations and the Mackey machine . . . . . . . . . . . . . . . 16

1.4.1 Induced representations . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.2 The Mackey Machine . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Square-integrable representations . . . . . . . . . . . . . . . . . . . . . . . . 191.6 The Fourier and Plancherel transformations . . . . . . . . . . . . . . . . . . 20

1.6.1 The Fourier algebra and Plancherel inversion . . . . . . . . . . . . . 24

2 Quantization on locally compact groups of type I 262.1 The quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2 Left and right quantizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3 Some operators arrising from the calculus . . . . . . . . . . . . . . . . . . . 27

2.3.1 Other convolution operators that appear in the literature . . . . . . 292.4 Quantization by a Weyl system . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Pseudo-differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.1 Relations between different τ -quantizations . . . . . . . . . . . . . . 362.6 Involutive algebras of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Some concrete examples of non-unimodular groups 383.1 A pseudo-differential calculus for the affine group . . . . . . . . . . . . . . . 38

3.1.1 Representation theory of the Affine group . . . . . . . . . . . . . . . 393.1.2 The quantization for the affine group . . . . . . . . . . . . . . . . . . 413.1.3 Operators that arise from the representation . . . . . . . . . . . . . 41

3.2 Grélaud’s group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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3.2.1 Representation theory of Gθ . . . . . . . . . . . . . . . . . . . . . . . 433.2.2 Computing the unitary dual . . . . . . . . . . . . . . . . . . . . . . . 443.2.3 The quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Crossed products of C∗-algebras 474.1 C∗-dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 The Schrödinger representation . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Interlude on Amenable groups . . . . . . . . . . . . . . . . . . . . . 50

5 Conclusions 51

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Introduction

M. Măntoiu and M. Ruzhansky developed [33] a global pseudo-differential calculus, orquantization, for the class of second countable locally compact unimodular type I groups.Our aim is to generalize their main results to the more general class of non-unimodulargroups. There are many important examples of non-unimodular groups, the simplest oneis perhaps the affine group consisting of all affine transformations of the real line, the onlynon-unimodular Lie group in dimension two. In dimension three there are many infinitefamilies of non-isomorphic non-unimodular Lie groups. Many other examples arise in thestudy of parabolic subgroups of semisimple Lie groups that are used to study irreduciblerepresentations through extensions of Mackey’s machine [19–23].

Let G be a locally compact group. It will be assumed that G is second countable andof type I. Let G be its unitary dual, that is, the space of all classes of unitary equivalenceof (strongly continuous) irreducible unitary representations. The formula (cf. eq. (1.1) [33]for the simpler unimodular case)

[Op(a)u] (x) =∫G

∫G

Tr(a(x, ξ)D

12ξ πξ(xy

−1)∗)

∆(y)−12u(y) dξ dy, (0.1)

is the starting point for a global pseudo-differential calculus on G. It involves specialmeasures on G and G, namely, the Haar and Plancherel measures, operator-valued sym-bols defined on G × G, the modular function ∆ of the group and a family of unboundedoperators; the formal dimension operators Dξ introduced by Duflo and Moore in [8, §3].Formula (0.1) also makes use of the Haar and Plancherel measures on G and G respec-tively. In order to make sense of formula (0.1), we also have to fix a measurable fieldof irreducible representations (πξ)ξ∈G such that πξ is a representations in the class of ξthat acts on a Hilbert space Hξ. For the moment the symbols are essentially chosen sothat the compositions a(x, ξ)D1/2

ξ are trace-class operators on the Hilbert space Hξ almosteverywhere. We also require the map sending ξ to the trace-class norm of a(x, ξ)D1/2

ξ tobe absolutely integrable for almost all x ∈ G.

The notion of quantization comes from the passage from classical mechanics to quantum

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mechanics. It is a rigorous formalism in which one passes from abelian C∗-algebras ofobservables, as in Hamiltonian mechanics, to non-abelian ones, as in quantum mechanicswhere the observables are operators on an infinite-dimensional Hilbert space. Quantizationshave been proven useful in the study of partial differential equations, quantum optics andsignal processing. It also has many connections to Lie theory, as it is directly connected[11] to the Heisenberg group and to the metaplectic group, which is the double covering ofthe symplectic group.

Formula (0.1) is a generalization of the one derived in [33, eq. (1.1)] to the class ofunimodular groups, but our formula has a difference in the order of the factors that hasto do with the choice of a convention for the Fourier transform (cf. Remark 1.4) ourquantization will give rise to right-invariant operators whereas the one in [33] gives rise toleft-invariant operators.

Particular cases of compact Lie groups have been extensively studied in [35, 36] forexample, and in the references cited therein. The class of nilpotent Lie groups is treatedin [10] and in other references. For a general treatment of pseudo-differential operators ina group-theoretic setting see [10, 35]. The idea of using the irreducible representations ofa group to define such calculus seems to come from [41, §1.2], but it was not developedin this abstract setting. All the articles cited above contain historical background andreferences to the existing literature treating pseudo-differential operators and quantizationin a group-theoretic context. For a historical survey on harmonic analysis see [32], forexample.

One of the advantages of using operator-valued symbols is that one gets a global ap-proach. Even for compact Lie groups there is no notion of full scalar-valued symbols for apseudo-differential operator using local coordinates. For a more detailed discussion of theadvantages of this approach see [33].

When our group is G = Rn, formula (0.1) boils down to the extensively studied Kohn-Nirenberg quantization rule. In that case there is a much bigger class of symbols, namelythe Hörmander symbol classes Smρ,δ(Rn). More general classes of symbols have been studied,but definitively the Hörmander classes are the most important ones, they are extensivelystudied in –but not only– [25,40,43], and the spectral theory of pseudo-differential operatorsis studied in [37]. For G = Rn there are also τ -quantizations given by

[Opτ (a)u](x) =∫Rn

∫Rna ((1− τ)x+ τy, ξ) e2πi〈ξ,x−y〉u(y) dξ dy, (0.2)

with τ ∈ [0, 1] mostly related to ordering issues. The Kohn-Nirenberg case amounts totake τ = 0. Another interesting case is τ = 1/2, the so-called Weyl quantization, whichis a more symmetric quantization that has the desirable property Op(a) = Op(a)∗. It is

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possible to extend the idea of τ -quantizations to our pseudo-differential calculus on typeI groups with a fixed measurable function τ : G → G instead of a real number τ ∈ [0, 1].For a τ -quantization the right modification of formula (0.1) turns out to be

[Opτ (a)u](x) =∫G

∫G

Tr(a(τ(yx−1)x, ξ)D

12ξ πξ(xy

−1)∗)

∆(y)−12u(y) dξ dy, (0.3)

from which one recovers (0.1) after setting τ(·) = e to be the constant function, where e isthe identity element of the group. Another interesting example is when τ(x) = x. In thesimplest case of G = Rn this amounts to taking τ = 1 and one gets the quantization inwhich derivatives are composed to the left and position operators to the right. Anyhow,the formalisms corresponding to different choices of τ are actually isomorphic when werestrict to the classes of symbols we are considering in the present thesis.

If G = Rn, then one can write the quantization as

Op(a) =∫Ga(ξ, x)W (ξ, x) dx dξ,

where the Weyl system is the family

W (ξ, x) = V (ξ)U(x) | (ξ, x) ∈ R2n,

of unitary operators in L2(Rn) obtained by putting together the operators of translationand modulation. The Weyl system offers a precise way to codify the canonical commutationrelations between position and momentum operators from quantum mechanics, and thequantization Op can be seen as a non-commutative functional calculus on these operators.Besides the physical interest, this opens the way to some new topics or tools such as theBargmann transform, the anti-Wick quantization, co-orbit spaces, and others. In Section2.4 we show how to carry this point of view to the general category of locally compacttype I groups, but one of the drawbacks of non-unimodular groups is that the resultingoperators are only defined in a dense subspace.

Weyl systems were one of the first examples of projective representations. The studyof projective representations of R2n was one of the most important problems in the 20thcentury and had its roots deep in the foundation of quantum mechanics. One can seethat the projective representations of R2n can be seen as unitary representations of theHeinsenberg group with n degrees of freedom [1] and the representations of the latterwere settled down with the Stone-von Neumann theorem, which says that under somehypothesis, and up to equivalence, there is only one possible representation that satisfiesthe canonical commutation relations.

Another approach to a quantization consists of using the formalism of C∗-algebras.

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Given a locally compact group G there is an action by left (or right) translations onvarious C∗-algebras of functions on G. In such situations there are natural crossed productsassociated to them (cf. Chapter 4): Among the non-degenerate representations of theseC∗-algebras we mention the Schrödinger representation, acting on the Hilbert space L2(G).This formalism allows us to take full advantage of the theory of C∗-algebras, extending tothe bigger class of compact operators on L2(G).

In the present thesis we are not going to rely on properties such as compactness,semisimplicity, nilpotency or smoothness. Almost all hypotheses shall be on the measure-theoretic side. The category of second-countable type I locally compact groups has a niceintegration theory and their unitary duals have an amenable integration theory. Thisframework allows for a general form of Plancherel’s Theorem, which is all that is neededto develop the basic features of a quantization, even for non-unimodular groups. The non-unimodular Plancherel theorem is originally due to [39] and had many contributions byDuflo, Moore [8], Lipsman and Kleppner [27], To apply the theorem one needs to knowthe complete unitary dual of a group, including its Plancherel measure. Later H. Führ [15]found the exact domain in which the inversion formula holds in the non-unimodular case.For an introduction to abstract harmonic analysis we refer to [12].

We now summarize the present thesis.

• In Chapter 1 we introduce notation used throughout the thesis and the general theoryand tools required to properly develop a quantization, including tools from abstractharmonic analysis and functional analysis, the main tool being the non-unimodularPlancherel transform.

• In Chapter 2 we make a preliminary construction of the quantization Op on a denselydefined subspace using formula (0.1). We include a discussion on the differencesbetween the left and right quantizations which comes from the non-commutativity ofthe group, and we study how to recover the families of convolution and multiplicationoperators using our quantization.

• In Section 2.4 we introduce the notion of a Weyl system, a measurable family ofdensely defined closed operators. Then we define a τ -quantization for an arbitrarymeasurable function τ : G→ G that has to do with ordering issues of the operators.In Section 2.5 we introduce a more general τ -quantization, and prove that it is aunitary map from our class of symbols onto the Hilbert-Schmidt operators on L2(G).

• In Sections 3.1 and 3.2 we work out the explicit formulas of the quantization forthe group of affine transformations of R and Grélaud’s group, two examples of non-unimodular solvable Lie groups. We compute the unitary dual of Grélaud’s group

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using the Mackey machine reviewed in § 1.4.

• In Chapter 4 we review the basic theory of crossed products of C∗-algebras and weshow how it relates to our theory. This formalism is also used to cover a bigger classof compact operators using the Schrödinger representation associated to a naturalcrossed product.

In the future it is our goal to extend formula (0.1) for more general classes of denselydefined operator-valued symbols, to cover for example differential operators on Lie groups,or even the class of bounded operators. Many developments have been done in this directionfor connected nilpotent Lie groups [10] and compact groups [36].

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Chapter 1

Framework

In this Chapter we set up the general framework of this thesis,and also recall some knownresults in the Fourier theory of non-unimodular groups of type I. We also briefly discuss thetheory of square-integrable representations, for which we have many explicit constructions.We also review the basic aspects of the Mackey machine that will be used to compute theunitary dual of Grélaud’s group in Section 3.1.

1.1 General remarks from functional analysis and measuretheory

We denote Hilbert spaces, over the field of complex numbers, with the letter H, using theconvention that their scalar product, denoted by 〈·, ·〉H, will be linear in the first variableand anti-linear in its second. In the following we assume all Hilbert spaces to be separable.If H is a Hilbert space we denote its conjugate Hilbert space by H†, whose elements are thesame as those of H but the scalar product is defined as α · u = αu, and its inner productis conjugate to the one from H, i.e. 〈u, v〉H† = 〈v, u〉H. B(H) denotes the C∗-algebra ofall bounded linear operators on H, and B0(H) stands for the two sided ∗-ideal of compactoperators on H. We also make use of the Schatten-von Neumann classes Bp(H) for p ≥ 1;these are Banach ∗-algebras with the norm

‖T‖Bp = Tr((T ∗T )p/2

)1/p.

The most important cases are B1(H), the space of trace-class operators, and B2(H), thespace of Hilbert-Schmidt operators. The latter when endowed with the inner product

〈T, S〉B2 = Tr (TS∗) ,

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becomes a H∗-algebra. As a Hilbert space it is unitarily isomorphic with the Hilbert tensorproduct H⊗H† in a natural way. Both B1(H) and B2(H) are two-sided ∗-ideals in B(H)whose closure in the operator norm is B0(H).

Let A be a set of bounded operators on a Hilbert space H. By A′ we denote the familyof all bounded operators on H that commute with all the elements of A. This set is a vonNeumann algebra, i.e. a C∗-algebra which is closed in the strong operator topology. Thereis a celebrated theorem due to von Neumann [4, Chapter IX, Theorem 6.4] that says thatif A is a C∗-subalgebra of B(H) containing the identity, then A′′ coincides with the closureof A in the weak operator topology.

A Borel space is a set X endowed with a σ-algebra of subsets of X, called Borel setsor measurable sets. We are going to refer interchangeably to Borel space as measurablespaces.

Definition 1.1. A measurable space X is called countably separated if there is acountable family Ωii∈N of measurable sets such that for all x ∈ X one has x = ⋂

x∈Ωi Ωi.The space X is called a standard Borel space if there is a measurable isomorphismX → Y , where Y is a complete second countable metric space, endowed with the Borelσ-algebra generated by its topology.

Clearly being a standard Borel space implies being countably generated, and the formeris a much stronger hypothesis that one may initially think. In a famous classification dueto Kuratowsky it is shown that every standard Borel space is Borel isomorphic either tothe interval [0, 1] or to a countable discrete set [34, Theorem 2.14]. We say that a measureυ is countably separated (or standard) if there is a measurable υ-conull subset Ω ⊆ Xthat is countably separated (standard).

Much of the interest of studying these properties comes from the study of quotientsof measurable spaces. Let ∼ be an equivalence relation on a measurable space X. Thequotient Borel structure on X/∼ is the finest σ-algebra making the natural projectionX → X/ ∼ a measurable map. Sometimes we shall need measurable transversalsΩ ⊆ X, that is a measurable set that contains exactly one element of each equivalenceclass in X/∼. Another construction comes as follows. Let X,Y be Borel spaces, and letf : X → Y be a Borel map, let υ be a given measure in X. We say that a measure υ on Yis a pseudo-image of υ, if υ is equivalent to the measure υf given on measurable sets byυf (Ω) = υ(f−1(Ω)), this means that for any measurable set Ω ⊆ Y one has that υ(Ω) = 0if and only if υ(f−1(Ω)) = 0.

Definition 1.2. Two measures υ1, υ2 on a Borel space X are called strongly equivalentif there is a continuous function g : X → (0,∞) such that for each compactly supported

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continuous function f : X → C one has∫Xf(x) dυ1(x) =

∫Xf(x)g(x) dυ2(x).

This does not implies equivalent, since the function g does not need to be integrable.

1.2 Direct integrals of Hilbert spaces

Let X be a Borel space. A field of Hilbert spaces over X is just a family (Hx)x∈X ofseparable Hilbert spaces Hx. A function s : X → ∐

x∈X Hx is called a section over X,or a vector field if sx ∈ Hx for all x ∈ X. A measurable field of Hilbert spacesover a field X is a field of Hilbert spaces (Hx)x∈X together with a countable set ej∞j=1of sections over X with the following properties:

(i) The functions x 7→ 〈eix, ejx〉 are measurable for all i, j ∈ N.

(ii) for each x ∈ X, the set eixi∈N is a total subset of Hx.

We say that a section s over X is a measurable section if the map x 7→ 〈sx, eix〉 ismeasurable for all i ∈ N.

Definition 1.3. Let X be a Borel space and let (Hx)x∈X be a measurable field of Hilbertspaces over X. Suppose that υ is a measure on X. The direct integral of the spaces Hx,denoted by ∫ ⊕

XHx dυ(x),

is the Hilbert space consisting the all the measurable sections s over X such that

‖s‖2 =∫X‖sx‖2Hx dυ(x) <∞,

in where two sections are identified if they coincide in a υ-conull set. The inner productof two sections s, s′ is given by

〈s, s′〉 =∫X〈sx, s′x〉Hx dυ(x).

Example 1.1. Let (X, ν) be a measure space. Then, if Hx = H is a fixed Hilbert spacefor all x ∈ X, then ∫ ⊕

XHx dυ(x) ∼= L2(X, υ;H).

This space is also naturally isomorphic to the Hilbert tensor product H⊗ L2(X, υ) underthe identification (η ⊗ u)(x) = u(x) η.

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Definition 1.4. A measurable field of operators (Tx)x∈X over X is a family of (notnecessarily bounded) operators Tx : Hx → Hx, where (Hx)x∈X is a measurable field ofHilbert spaces, such that the section (Txsx)x∈X is measurable whenever s is a measurablesection over X.

Suppose that υ is a measure on a Borel space X, then if T = (Tx)x∈X is a measurablefield of operators such that ess supx∈X‖Tx‖ is finite, then T defines a bounded operator onthe direct integral

∫⊕X Hx dυ(x) with operator norm bounded by the essential supremum of

the norms ‖Tx‖; we denote this operator by([∫ ⊕XTy dυ(y)

]s

)x

= Txsx.

1.3 Basic representation theory

Let G be a Hausdorff second countable locally compact group. For the most part thismeans that the product and inversion laws of the group are continuous maps, and as atopological space G is both second countable and locally compact. We denote its unit bye ∈ G.

Remark 1.1. Recall that a second countable group is separable, Hausdorff, σ-compact andcompletely metrizable. In particular, as a measurable space it will be standard. Sinceevery irreducible representation is cyclic, it must act on a separable Hilbert space.

Let H be a closed subgroup of G. Then the quotient G/H, endowed with the quotienttopology, is a locally compact topological space. It is even a locally compact group whenH is a normal subgroup. The spaces G and G/H are completely metrizable. When theyare endowed with the Borel σ-algebra generated by their topology they become standardBorel spaces. The following proposition (cf. [28] Lemma 1.1) is due to G. Mackey.

Proposition 1.1. Let G be a second countable group and let H be a closed subgroup. Thenthere exist a measurable transversal of G/H.

Definition 1.5. A representation of a group G on a Hilbert space Hπ is a functionπ : G→ B(Hπ), such that for each pair of elements x, y ∈ G

π(xy) = π(x)π(y).

The representation is called unitary if π(x)∗ = π(x−1) for all x ∈ G. It is called a stronglycontinuous representation if for each u ∈ Hπ the map x 7→ π(x)u is continuous. Everyunitary representation considered in this thesis is strongly continuous, even if it is notexplicitly mentioned.

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Definition 1.6. A projective representation on a Hilbert space Hπ is a map π : G→Hπ such that there exist a measurable map ω : G×G→ S1 into the unit circle satisfying

π(x)π(y) = ω(x, y)π(xy),

and that for each pair of vectors u, v ∈ Hπ, the map x 7→ 〈u, π(x)v〉 is measurable. Wesay that π is a projective representation with multiplier ω, or that it is a ω-projectiverepresentation.

Definition 1.7. A representation π on a Hilbert space Hπ is called irreducible if thereare no proper closed linear subspaces E ⊆ Hπ such that π(x)E ⊆ E for all x ∈ G.

Definition 1.8. Let π, σ be two representations. An intertwining operator T : Hπ →Hσ is a bounded operator such that for each x ∈ G one has

T π(x) = σ(x)T.

We say that two representations are unitarily equivalent if there exist an unitary inter-twining operator between them.

The most basic, and perhaps, the most important result about irreducible representa-tions is the following.

Proposition 1.2 (Schur’s lemma). A unitary (projective) representation π is irreducibleif and only if π(G)′ = C · IdHπ . Suppose that π1, π2 are irreducible unitary (projective)representations of G. If they are equivalent, then there exists a unique (up to multiplica-tion by a constant) intertwining operator. Otherwise there are no non-trivial intertwiningoperators.

Let π be a strongly continuous unitary representation of G on a Hilbert space Hπ. Forany such representation we also have its contragradient π† acting on Hπ† by π†(x)u =π(x)u. Note that in general π is not equivalent to π†.

We say that π′ is a subrepresentation of π if it is unitarily equivalent to the restrictionπ|E , where E is a closed invariant subspace of Hπ.

Definition 1.9. We say that two representations π, ρ are quasi-equivalent if for everysubrepresentation π′ of π there is a non-trivial operator intertwining π′ and ρ, and for everysubrepresentation ρ′ of ρ there is a non-trivial operator intertwining ρ′ and π. This is anequivalence relation among representations. Two equivalent representations are clearlyquasi-equivalent and the notions coincide when we refer to irreducible representations.

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Definition 1.10. A unitary representation π is called primary, or a factor represen-tation, if the center of the von Neumann algebra generated by π(G) ⊆ B(Hπ) consist onlyof multiples of the identity i.e.

π(G)′ ∩ π(G)′′ = C · IdHπ .

Definition 1.11. A unitary representation π of G is called multiplicity-free if the vonNeummann algebra π(G)′ is commutative. It is called a type I representation if it isquasi-equivalent to a multiplicity-free representation.

By Schur’s lemma one has that any irreducible representation is a multiplicity-freeprimary representation, but the converse is not necessarily true, this is the content of thefollowing definitions.

Definition 1.12. We say that a topological group G is type I if every primary representa-tion is quasi-equivalent to an irreducible representation, or equivalently, if it is a direct sumof copies of some irreducible representation. If G is second countable, another characteri-zation is that the group is type I if and only if every representation is type I [15, Theorem3.23].

Example 1.2. Some examples of type I groups are:

– Compact groups.– Connected semisimple Lie groups [20].– Abelian groups.– Exponentially solvable Lie groups [2], in particular connected simply connected nilpo-tent Lie groups.

– Real algebraic groups [6].

It is known that a discrete group is of type I if and only if it possesses an abelian normalsubgroup of finite index [42].

Fix a left Haar measure µ on G, that is, a Radon measure µ such that∫Gf(xy) dµ(y) =

∫Gf(y) dµ(y),

for all f ∈ Cc(G) and x ∈ G. We will denote this choice of left Haar measure by dµ(x) =dx; Every locally compact groups admits a left Haar measure, and it is unique up to apositive constant. Once µ is fixed we get a right Haar measure µr defined by the formulaµr(Ω) = µ(Ω−1).

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Let ∆ : G→ (0,∞) be the modular function of G, defined by the formula µ(Ωx) =∆(x)µ(Ω) for all measurable sets Ω ⊆ G and x ∈ G. This implies in particular thatdµr = ∆−1 dµ. Hence the left and right Haar measures are strongly equivalent.

The modular function is a continuous (smooth if G is a Lie group) homomorphisminto the multiplicative group (0,∞). We say that a group is unimodular if the modularfunction is a constant function. If G is a connected Lie group then

∆(x) = |det Ad(x−1)|,

where Ad is the adjoint representation of G on its Lie algebra [24, Chapter 10, Lemma1.2].

Remark 1.2. The the following classes of groups are unimodular:

– Connected semisimple Lie groups.– Abelian groups.– Connected nilpotent Lie groups.– Compact groups.– Discrete groups.

We also note that G is unimodular provided that the abelianization of G is a compactgroup.

Let N be the kernel of the modular function; it is a closed normal subgroup of G, whichis itself a unimodular group [12, Theorem 2.49]. We also note that since our groups arelocally compact the image of the modular function is a subgroup of R>0. It may be eitherdense in (0,∞) or a closed discrete subset.

The spaces Lp(G) = Lp(G,µ) of p-integrable complex-valued functions on G will alwaysrefer to the left Haar measure. These are separable Banach spaces for p ∈ [1,∞). By Cc(G)we denote the space of continuous complex-valued functions on G with compact support,a dense subspace of Lp(G). The space C0(G) denotes the C∗-algebra of all continuouscomplex-valued functions defined on G that vanish at infinity.

1.3.1 The left and right regular representations

Every group naturally comes with a pair of unitary representations, namely, the left andright regular representations, defined on L2(G) by

λyf(x) = f(y−1x)

ρyf(x) = ∆(y)12 f(xy).

12

These are unitary strongly continuous representations of G [12, §2.5]. It is a deep fact [38]that one has the following equality of von Neumann algebras

λ(G)′′ = ρ(G)′, ρ(G)′′ = λ(G)′.

There is also the two-sided regular representation λ⊗ ρ of G×G given by

λ⊗ ρ (x, y)f = λxρyf f ∈ L2(G).

For the convenience of the reader we recall that the modular function plays the followingrole in integration by substitution of variables∫

Gf(y) dy =

∫G

∆(x)f(yx) dy =∫G

∆(y)−1f(y−1) dy. (1.1)

The modular function implements a Banach ∗-algebra structure on L1(G): the convolutionof two functions defined by the integral

(f ∗ g)(x) =∫Gf(y)g(y−1x) dy,

and the involution is given by

f∗(x) = ∆(x)−1f(x−1).

In general, one has a p-dependent involution on Lp(G) given by

f∗(x) = ∆(x)−1p f(x−1). (1.2)

In the following we reserve the notation f∗ for functions in the Hilbert space L2(G).

Definition 1.13. Let X be a Borel space. A measurable field of representationsof G over X is a measurable field of operators (πy)y∈X such that each πy is a unitaryrepresentation of G. A measurable field of representations induces a unitary representationon the direct integral

∫⊕X Hy dυ(y) given on sections s over X by

([∫ ⊕Xπy(x) dυ(y)

]s

)p

= πp(x)sp.

An important result due to G. Mackey [30, Theorem 10.2] is that for each measurablesubset Ω ⊆ G on which the Mackey Borel structure is standard, there exist a measurablefield of irreducible representations (πξ)ξ∈Ω over Ω, acting on canonical Hilbert spaces Hξ,such that πξ ∈ ξ for each ξ ∈ Ω.

13

Consider the Banach ∗-algebra L1(G) endowed with the universal norm

‖f‖C∗(G) = supρ‖ρ(f)‖,

where the supremum is taken over the set of all non-degenerate ∗-representations of L1(G).The completion of L1(G) with respect to the universal norm is called the group algebraof G. We denote this C∗-algebra as C∗(G). It is a standard fact that the irreducibleunitary representations of G are in one-to-one correspondence with the non-degenerate∗-representations of C∗(G), the correspondence being given by

π(f) =∫Gf(y)π(y) dy

for functions in the dense subspace L1(G).Remark 1.3. For a type I group G, its C∗-enveloping algebra is postliminal. An equivalentformulation of the type I hypothesis is that for all irreducible representations π one hasthat π(C∗(G)) contains all the compact operators on Hπ.

Given a locally compact group G, its unitary dual G is the collection of all of itsirreducible unitary representation modulo unitary equivalence. For a representative πξ ∈ ξof an element of the unitary dual of G, we denote the Hilbert space on which it acts byHξ = Hπξ . The unitary dual is known completely for various classes of groups, includingabelian, compact, and connected nilpotent Lie groups. It is known up to a set of measurezero in the case of connected semisimple Lie groups. In the case of abelian groups, G isalso a second countable locally compact group, but as soon one leaves the abelian world,there does not seem to be a natural way in which this space is a group, even for the caseof compact groups.

We endow G with the Mackey Borel structure introduced in [29, §9]. This is done asfollows: Let Irrn(G) be the set of all irreducible unitary representation acting on a fixedHilbert space Hn of dimension n ∈ N ∪ ∞ and let Irr(G) = ⋃∞

n=0 Irrn(G). We endoweach Irrn(G) with the weakest σ-algebra such that the maps π 7→ 〈u, π(x)v〉 are measurablefor all u, v ∈ Hn, x ∈ G. Then a set Ω ⊆ Irr(G) is said to be measurable if and only ifΩ ∩ Irrn(G) is measurable for all n ∈ N ∪ ∞. The Mackey Borel structure is thequotient Borel structure induced by the map Irr(G) → G that sends each representationto its equivalence class. The next proposition (cf. [15] Lemma 3.15) sheds some light onthe Mackey Borel structure.

Proposition 1.3. Assume that X is a standard Borel space and let (πx)x∈X be a mea-surable field of irreducible representations of G. Then the map X 3 x 7→ [πx], where [π]denotes the equivalence class of π, is a measurable map into G.

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A consequence of G being of type I is that G is a standard Borel space [18]. It isknown [7] that being of type I is equivalent to G being countably separated and is alsoequivalent to being a standard Borel space. Since spaces which does not satisfy the formerhypothesis are badly behaved, it is very hard to expect to have a reasonable integrationtheory for non-type I groups.

One also provides G with the Fell topology. This topology is T0 provided that G is oftype I, and then the Mackey Borel structure coincides with the Borel σ-algebra generatedby this topology. For semisimple and nilpotent connected Lie groups this topology is T1,but in general this space is not Hausdorff. For a proof of these assertions see [7, 18], andfor more information on the Fell topology see [9, Chapter VII §1].

Definition 1.14. We say that a standard measure ν on G is a Plancherel measure(cf. [15] Definition 3.30 and Theorem 3.24) if it yields a direct integral central decompositionof the left regular representations into irreducible representations. That is to say, ν is aPlancherel measure if the following conditions hold

(i) There is a measurable map m : G → N ∪ 0,∞, a measurable field of irreduciblerepresentations (πξ)ξ∈G with πξ ∈ ξ, and a unitary isomorphism U : L2(G) →∫⊕Gm(ξ) · Hξ dν(ξ) such that such that for each element x ∈ G one has

Uλx =∫ ⊕Gm(ξ) · πξ(x) dν(ξ)U.

More precisely, let In be a set with n elements endowed with the counting measure,then m(ξ) · Hξ = L2(Im(ξ))⊗Hξ and m(ξ) · πξ(x) = IdL2(Im(ξ)) ⊗ πξ(x).

(ii) U implements an equivalence of von Neumann algebras

λ(G)′ ∩ λ(G)′′ ∼=∫ ⊕G

C · Idm(ξ)·Hξ dν(ξ).

Plancherel measures do exist for separable locally compact groups of type I and infact they are all mutually equivalent (cf. Theorem 1.3 below). From now on we adopt thenotation dν(ξ) = dξ for a Plancherel measure ν. If necessary, we will denote by νG thePlancherel measure of a group G.

There are various cases in which the Plancherel measure can be given explicitly. Forabelian groups their unitary dual is also an abelian group in a canonical way and thePlancherel measure coincides with a multiple of its Haar measure. For connected simplyconnected nilpotent Lie groups it corresponds to a measure on the space of coadjoint orbitsarising from the Lebesgue measure on g∗ [26, Chapter 3 §2.7]. For compact groups thePeter-Weyl theorem says that the irreducible representations form a discrete set and that

15

the Plancherel measure of an irreducible representation is equal to 1. Note that this is onlyvalid using our convention of the Plancherel transform in which the Duflo-Moore operatorsare taken into account (cf. Theorem 1.2). For a proof see [12, Theorem 5.12].

1.4 Induced representations and the Mackey machine

The Mackey machine is one of the most important tools for computing the unitary dualof a general locally compact group, it consist of inducing representations from a closednormal subgroup and a family of “small subgroups” that appear in the quotient group.This method is developed mainly in [30] and is the basis of a more geometric study ofthe unitary dual. This point has been proven to be very fruitful, it is directly connectedto Kirillov’s orbit method, which is the tool of choice to compute the unitary dual of aconnected simply connected nilpotent Lie group. This method can be even extended toexponentially solvable Lie groups without many changes. It is even possible to extend thismethod for the class of connected semisimple Lie groups, since they do not have properclosed normal subgroups, one has to study the parabolic subgroups instead, and there is asimilar result in this case allowing one to compute at least the support of the Plancherelmeasure.

1.4.1 Induced representations

Let υ be a Borel measure on a locally compact Hausdorff G-space X, define the measuresυx(Ω) = υ(x−1Ω) for each Borel subset Ω ⊆ X, and all x ∈ G. Note that for all integrablefunctions f one has ∫

Xf(p) dυx(p) =

∫Xf(xp) dυ(p). (1.3)

Definition 1.15. A measure υ on a G-space X is said to be invariant if the measuresυx coincide. It is called quasi-invariant if the measures υx are all mutually absolutelycontinuous. We will call υ strongly quasi-invariant if the Radon-Nikodym derivative(x, p) 7→ (dυx/dυ)(p) is a continuous function defined on G × X. This is a strongerassumption than the hypothesis of having all the measures υx mutually strongly equivalent,since the Radon-Nikodym derivative must be jointly continuous.

It is a standard fact [12, §2.6] that any transitive locally compact G-space admits astrongly quasi-invariant measure and in fact it is unique up to strong equivalence. Moreoverif X = G/H is a transitive G-space, there is a G-invariant Radon measure υ on X if andonly if the modular functions on G and H satisfy the relation ∆G

∣∣H

= ∆H . In such a case,υ unique up to a positive factor.

16

Let H be a closed subgroup of G, and assume that X = G/H admits a strongly quasi-equivalent measure υ. Let σ be a representation of H on a Hilbert space Hσ. The inducedrepresentation π = IndGH(σ) acts on the Hilbert space Hπ = L2(G,H, σ, υ) consisting ofclasses of equivalence of functions f : G→ Hσ such that

(i) The maps x 7→ 〈f(x), u〉 are measurable for all u ∈ Hσ.

(ii) f(xh) = σ(h)∗f(x) for all x ∈ G, h ∈ H, except for a possible set of pairs (x, h) suchthat the corresponding products xh’s belong to a υ-null set in G/H.

(iii) The following quantity is finite

‖f‖2 =∫G/H‖f(x)‖2 dυ(xH).

As usual, we impose the equivalence relation f ∼= g if and only if ‖f − g‖ = 0 (cf. [16] §4,Theorem 9). The inner product of Hπ is given by

〈f, g〉 =∫G/H〈f(x), g(x)〉Hσ dυ(xH).

Thanks to condition (ii) the quantities 〈f(x), g(x)〉Hσ depends only on the right H-cosetof x, so the formulas above are well defined. The induced representation π is given byformula

(π(x)f) (y) =√dυxdυ

(yH) f(x−1y).

Thanks to formula (1.3), and the fact that the Radon-Nikodym are jointly continuous, πis a strongly continuous unitary representation of G.

1.4.2 The Mackey Machine

Let N be a closed normal subgroup of G. The dual action of G on the unitary dual of Nis given by

x.σ(n) = σ(x−1nx) x ∈ G,n ∈ N.

With this action N becomes a Borel G-space. We denote by Gσ the stabilizer in G ofσ ∈ N . Let H = G/N , since N acts trivially on N , the dual action gives rise to an actionof H. We denote by Hσ the stabilizer in H of σ ∈ N . These groups are usually called“small groups” in the literature. We denote the orbit of an element σ ∈ N as Oσ, the typeI hypothesis implies that Oσ ∼= N/Hσ as Borel spaces [15, §3.2].

Definition 1.16. Let N be a closed normal subgroup of G and let H = G/N , endowed

17

with the Borel quotient structure. We say that N is regularly embedded in G if theorbit space N/H is a countably separated Borel space.

Let N = ker(∆) and let H = G/N . It is proved in [8, Theorem 6] that the left regularrepresentation of G is type I if and only if the left regular representation of N is type I andthe orbit space N/H is a standard Borel space. In particular since our groups are type I,N is automatically regularly embedded in G.

Definition 1.17. We say that a representation σ ∈ N has trivial Mackey obstructionif it can be extended to a unitary representation of Gσ.

There are two very simple cases in which a representation σ ∈ N has trivial Mackeyobstruction. Namely when Gσ = N or when G = N oH is a semi-direct product and Nis abelian. Nevertheless, these cases cover a great number of examples.

In general it is possible to extend σ to a projective representation of Gσ in the followingway. If x ∈ Hσ stabilizes σ, there is a unitary operator Ux such that

Uxσ(n)U∗x = σ(xnx−1).

This choice is unique up to a factor of norm 1. Without loss of generality we choose themso that Ue is the identity and x 7→ Ux is a measurable map, that is to say, x 7→ 〈u, Uxv〉is a measurable map for all u, v ∈ Hσ. Let Ω be a measurable transversal of Gσ/N . Anelement y ∈ Gσ can be written in a unique way as y = sn, where s ∈ Ω and n ∈ N . Wedefine σ(y) = UsNσ(n). Then one sees that for all x, y ∈ Gσ

UxUyσ(n)U∗yU∗x = σ(xyn(xy)−1).

Hence, by the uniqueness of the operators Ux’s, there is a constant ω(x, y) of norm 1 suchthat UxUy = ω(x, y)Uxy for all x, y ∈ Gσ. Since U is measurable, one sees that ω ismeasurable. Hence σ is a projective representation that extends σ to Gσ with multiplierω. The Mackey obstruction is trivial precisely when we can chose σ so that ω = 1.

We cite the celebrated theorem of G. Mackey [30] which is the main tool for computingthe unitary dual of a group in terms of a normal subgroup N and a family of “smallgroups” that depend on the dual action. For simplicity we restrict ourselves to unitaryrepresentations. The hypothesis of having trivial Mackey obstruction can be removed, butthe resulting induced representations turn out to be projective representations instead ofunitary.

Theorem 1.1 (The Mackey machine). Let N ⊆ G be a regularly embedded closed normalsubgroup and suppose that each σ ∈ N has trivial Mackey obstruction. Given a representa-tion σ ∈ N , denote a fixed extension to Gσ by σ. Given a unitary representation ρ ∈ Hσ,

18

we define the unitary representation σ× ρ of Gσ, which acts on the Hilbert tensor productHσ ⊗Hρ by

(σ × ρ)(x) = σ(x)⊗ ρ(xN).

Then one has that

(i) The induced representation IndGGσ(σ×ρ) is a unitary irreducible representation of G.(ii) Suppose that N is of type I. Then every irreducible unitary representation of G is of

the above form.(iii) Moreover, if we fix a measurable transversal Ω ⊆ N i.e. a set that contains exactly

one representative for each orbit in N/H, then the map

(σ, ρ) | σ ∈ N , ρ ∈ Hσ → G given by (σ, ρ) 7→ IndGGσ(σ × ρ),

is a bijection.

We also cite an extension of the theorem due to Kleppner and Lipsman [27] that allowsus to compute the Plancherel measure of the unitary dual. And only requires the Mackeyobstruction to be trivial νN -almost everywhere.

Proposition 1.4. Let G be a locally compact group and let N be a closed normal type Iunimodular subgroup. Suppose that N is regularly embedded and that there is a G-invariantmeasurable νN -conull subset Ω ⊆ N such that all σ ∈ Ω have trivial Mackey obstruction.Then the set ⋃

Oσ∈Ω/HIndGGσ(σ × ρ) | ρ ∈ Hσ,

is a ν-conull subset of G and the Plancherel measure may be obtained as follows: Pick apseudo-image νN of the Plancherel measure of N on Ω/H. Then there is for νN -almostall Oσ ∈ Ω/H, a normalized Plancherel measure νHσ such that if we identify IndGGσ(σ× ρ)as the point (σ, ρ) one has that the Plancherel measure of G is given by

dν(σ, ρ) = dνHσ(ρ) dνN (Oσ).

1.5 Square-integrable representations

Given a representation π on a Hilbert space Hπ, and two vectors u, v ∈ Hπ, we define thematrix coefficient of π at the pair (u, v) as

Cu,v(x) = 〈u, π(x)v〉.

19

Note that Cu,v is a uniformly continuous bounded function and the algebra generated byall the matrix coefficients of a given representation depends only on its equivalence class. Ifa matrix coefficient Cu,v is square-integrable for some non-zero u, v ∈ Hπ we say that π is asquare-integrable representation. If π is an irreducible representation it is known [7] thatthe representation is square integrable if and only if Cu,v ∈ L2(G) holds for all u, v ∈ Hπ.

Theorem 1.2. Suppose π is a square integrable irreducible unitary representation. Thereis a densely defined positive self-adjoint operator Dπ : Dom(Dπ)→ Hπ with dense image,called the Duflo-Moore operator, satisfying:

(i) For all vectors u, u′ ∈ Hπ, v, v′ ∈ Dom(D1/2π )⟨

Cu,D

1/2π v

, Cu′,D

1/2π v′

⟩L2(G)

= 〈u, u′〉〈v′, v〉.

In particular this implements an isometric linear map C : Hπ ⊗H†π → L2(G).

(ii) Up to normalization by a positive constant, Dπ is uniquely determined by the relation

π(x)Dππ(x)∗ = ∆(x)−1Dπ. (1.4)

We presented this result as it is stated in [15, p. 97], in which an explicit constructionof the operators Dπ is made for square integrable representations. When the group isunimodular the operators Dπ are just multiplication by a positive scalar dπ that coincideswith the dimension of Hπ when the latter is finite. An operator satisfying (1.4) is calledsemi-invariant with weight ∆−1. In general not all irreducible representations admit asemi-invariant operator, they only exist ν-a.e. (cf. Theorem 1.3).

1.6 The Fourier and Plancherel transformations

Suppose we have fixed a Plancherel measure ν in G, a measurable field of representations(πξ)ξ∈G and there is a family of densely defined self-adjoint positive operators Dξ : Hξ →Hξ satisfying relation (1.4) for ν-almost all ξ ∈ G. We define (in the weak sense) theoperator-valued Fourier transform of a function f ∈ L1(G) as

F(f)(ξ) =∫Gf(y)πξ(y) dy.

This is the unique operator F(f)(ξ) such that for all u, v ∈ L2(G) one has

〈F(f)(ξ)u, v〉 =∫Gf(y)〈πξ(y)u, v〉 dy.

20

The Fourier transform is a non-degenerate ∗-representation of L1(G), but in the non-unimodular case it fails to intertwine the two-sided regular representation of G with

∫⊕Gξ⊗

ξ† dξ and it also fails to be a unitary map. So we also introduce the Plancherel transformof f ∈ L1(G) ∩ L2(G) as the operator

P(f)(ξ) = F(f)(ξ)D12ξ .

In the following we denote by P(f) = f , the Plancherel transform of a function f .The Plancherel transform satisfies the following identities for functions f, g ∈ L2(G) ∩

L1(G),

πξ(x)f(ξ)πξ(y)∗ = λxρyf(ξ),

f(ξ)∗ = f∗(ξ),

f ∗ g (ξ) = πξ(f)g(ξ).

When πξ is square integrable, f(ξ) extends to a Hilbert-Schmidt operator on Hξ. It turnsout that this will hold for ν-almost every ξ ∈ G and not only for the representations whichare square-integrable (cf. Theorem 1.3 below); for a proof see [8, Theorem 5.1].

We are going to present a formulation of the Plancherel Theorem for non-unimodulargroups. For a proof in the unimodular case we refer to [7]. The non-unimodular PlancherelTheorem was developed by N. Tatsuuma [39] and later an extension of his theory, includinga clarification of the role of the hypothesis required, was obtained by Duflo and Moore [8].Similar results were obtained by Kleppner and Lipsman in [27]. We state the followingtheorem as it was derived in the article of Duflo and Moore and in the spirit of [15, Theorem3.48].

Theorem 1.3. Let G be a type I second countable locally compact group. Then thereexists a σ-finite Plancherel measure ν on G, a measurable field of Hilbert spaces (Hξ)ξ∈G,a measurable field of irreducible representations (πξ)ξ∈G with πξ ∈ ξ, and a measurablefield (Dξ)ξ∈G of densely defined self-adjoint positive operators on Hξ with dense imagesatisfying (1.4) for ν-almost every ξ ∈ G, which have the following properties:

(i) Let f ∈ L1(G) ∩ L2(G). For ν-almost all ξ ∈ G, the operator f(ξ) extends to aHilbert-Schmidt operator on Hξ and

‖f‖22 =∫G‖f(ξ)‖2B2 dξ. (1.5)

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(ii) The Plancherel transformation extends in a unique way to a unitary operator

P : L2(G)→∫ ⊕GB2(Hξ) dξ. (1.6)

(iii) P implements the following unitary equivalences of representations and von Neumannalgebras

λx ∼=∫ ⊕Gπξ(x)⊗ IdH†

ξdξ, (1.7)

ρx ∼=∫ ⊕G

IdHξ ⊗ π†ξ(x) dξ, (1.8)

λ(G)′ ∼=∫ ⊕G

C · IdHξ ⊗ B(H†ξ) dξ, (1.9)

λ(G)′′ ∼=∫ ⊕GB(Hξ)⊗ C · IdH†

ξdξ. (1.10)

In particular these relations show that ν satisfies the axioms required by Definition1.14 to be a Plancherel measure.

(iv) The Plancherel measure and the operator field may be chosen to satisfy the inversionformula

f(x) =∫G

Tr(f(ξ)D

12ξ πξ(x)∗

)dξ, (1.11)

for all f in the Fourier algebra of G (cf. Section 1.6.1 below). The integral in theinversion formula converges absolutely in the sense that f(ξ)D

12ξ extends to a trace-

class operator ν-a.e. and the integral over G of the trace-class norms is finite.

(v) The choice of(ν, (Dξ)ξ∈G

)is essentially unique: The semi-invariance relation (1.4)

fixes each Dξ up to a multiplicative constant, and once we fix these, ν is fixed by (1.11).On the other hand if we fix ν, (which is unique up to equivalence) the operators Dξ

are completely determined by ν.

(vi) G is unimodular if and only if there exist positive constants dξ such that Dξ = dξ IdHξfor ν-almost all ξ. If G is non-unimodular, Dξ is an unbounded operator for ν-almostall ξ (this can be seen from equation (1.4)).

(vii) Suppose there is another Plancherel measure ν ′ on G and measurable fields (πξ ′, Dξ′)ξ∈G

that share the properties i)-iii). Then ν and ν ′ are equivalent measures, and there isa measurable field of unitary operators (Uξ)ξ∈G, intertwining πξ and πξ ′, such that

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for ν-almost all ξ ∈ G the Radon-Nikodym derivative of ν ′ with respect to ν satisfies

dν ′

dν(ξ)Dξ

′ = UξDξ U∗ξ . (1.12)

The operators Dξ are called the formal dimension operators, or the Duflo-Mooreoperators. When π is induced from a subgroup H on which the modular function istrivial, the Hilbert space Hπ is then formed by vector-valued functions on G, and theDuflo-Moore operators have the very simple expression (cf. [12] Theorem 7.42)

(Dπf)(x) = ∆(x)f(x).

Moreover, if we require for N to be type I, then ν-a.e. the representation ξ ∈ G is inducedfrom a representation of N (cf. [39]).

In the unimodular case, if one replaces the Plancherel transform with the usual Fouriertransform, then formula (1.11) reads

f(x) =∫Gdξ · Tr

(f(ξ)πξ(x)∗

)dξ.

Most books that treat the unimodular case refer to dξ · ν as the Plancherel measure.This explains how to recover the unimodular theory using the non-unimodular Planchereltheorem.

Remark 1.4. Alternatively, we could also define the Plancherel transform of a integrablefunction f as

P(f)(ξ) = D12ξ

∫Gf(x)πξ(x)∗ dx.

Using the semi-invariance relation of Dξ and the involution given in (1.2), we get thefollowing relation

P(f)(ξ) =∫Gf(x)πξ(x)∗D

12ξ ∆(x)−

12 dx

=∫Gf(x−1)∆(x)−

12πξ(x)D

12ξ dx

= P(f∗)(ξ),

so the two definitions differ by an automorphism of L2(G). Another thing to have in mind

23

is that the inversion formula (1.11) takes the form

f(x) = f∗(x−1)∆(x)−12

=∫G

Tr(P(f∗)(ξ)D

12ξ πξ(x

−1)∗)

∆(x)−12 dξ

=∫G

Tr(D

12ξ P(f)(ξ)πξ(x)

)dξ.

For simplicity we make use of the following notation

B⊕2 (G) =∫ ⊕GB2(Hξ) dξ, B⊕1 (G) =

∫ ⊕GB1(Hξ)D

− 12

ξ dξ,

S(G) = L2(G)⊗ B⊕2 (G), S(G) = B⊕2 (G)⊗ L2(G).

S(G) will be the natural space for our symbols. It comes with a natural inner productgiven by

〈a, b〉S(G) =∫G

∫G

Tr (a(x, ξ)b(x, ξ)∗) dξ dx.

Remark 1.5. By formula (1.7), a representation πξ ∈ ξ is subrepresentation of the leftregular representation if and only if the singleton ξ has positive Plancherel measure. Itis known that a representation appears as a summand in the decomposition of λ only ifit is square-integrable. In addition one checks that for unimodular groups, all the square-integrable representations satisfy ν(ξ) = 1 [15, pp. 84], for some normalization. Whenthe Hilbert space has finite dimension dπ, the normalization is given by Dπ = dπ · Id.

1.6.1 The Fourier algebra and Plancherel inversion

Most of the results of this section are presented in the works [14,15] of H. Führ. In order toshed some light on the trace-class hypothesis imposed on our symbols, we elaborate a littleon the natural domain of the Plancherel transform in such a way that formula (1.11) holds.We also give the natural domain on the Plancherel side for the inversion formula (1.11).

Definition 1.18. The Fourier algebra A(G) of a locally compact group G is defined asthe closure of the linear span of

f ∗ g[ | f, g ∈ L2(G),

where g[(x) = g(x−1), with the norm

‖u‖A(G) = inf‖f‖2‖g‖2 | u = f ∗ g[.

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It becomes a Banach ∗-algebra with convolution as the product law and [ as the involution.This is the space of matrix coefficient functions of the left regular representation of G(cf. (1.13) below).

We record some calculations for further use in the following lemma (cf. [15] Lemma4.14).

Lemma 1.4. Let f, g be two square integrable functions on G. Suppose that g[ ∈ L1(G),then if

h(x) = 〈f, λxg〉 = (f ∗ g[)(x). (1.13)

Is a matrix coefficient of λ, we have that

h(ξ) = f(ξ) g(ξ)∗D−12

ξ . (1.14)

Hence h(ξ)D12ξ extends to a trace-class operator ν-almost everywhere.

For a function in L2(G) ∩A(G), by the previous lemma, its Plancherel transform is inB⊕1 (G) ∩ B⊕2 (G) for ν-almost every ξ ∈ G. A straightforward calculation shows that theinversion formula holds for such a function.

In [15, Theorem 4.12] it is shown that the Plancherel transform induces an isomorphismbetween the Banach spaces A(G) and B⊕1 (G). This induces an isomorphism of Hilbertspaces

P : A(G) ∩ L2(G)→ B⊕1 (G) ∩ B⊕2 (G).

The next proposition (cf. [15] Theorem 4.15) shows that A(G)∩L2(G) is the natural domainof the Plancherel transform in such a way that the inversion formula holds. Consideringthe preceding paragraph it also shows that, on the Plancherel side, the natural domain forthe inversion formula is B⊕1 (G) ∩ B⊕2 (G).

Proposition 1.5. Let F ∈ B⊕2 (G) and suppose that for ν-almost everywhere the operatorF (ξ)D

12ξ extends to a trace-class operator. Suppose moreover that

∫G‖F (ξ)D

12ξ ‖B1 dξ <∞.

If f is the inverse Plancherel transform of F , then we have for µ-almost everywhere

f(x) =∫G

Tr(F (ξ)D

12ξ πξ(x)∗

)dξ. (1.15)

25

Chapter 2

Quantization on locally compactgroups of type I

In this chapter we introduce a quantization leading to a pseudo-differential calculus foroperator-valued symbols defined on the whole group times its dual. In order to do so wetake advantage of the irreducible representations of the group. In section 2.4 we developthe notion of a Weyl system which will be used to make a sense of the formulas in a rigorousway for more general symbols. This will also makes things clearer when dealing with ageneral τ -quantization.

In the following we fix a Plancherel measure and choice of a measurable field of ir-reducible representations (πξ)ξ∈G and formal dimension operators (Dξ)ξ∈G so that The-orem 1.3 holds. By (vii) of Theorem 1.3, different choices of the measurable fields ofrepresentation or formal dimension operators will lead to isomorphic formulations.

2.1 The quantization

Given a symbol a ∈ L2(G) ⊗ (B⊕1 (G) ∩ B⊕2 (G)) we define the operator Op(a) : L2(G) →L2(G) with symbol a to be

[Op(a)u] (x) =∫G

∫G

Tr(a(x, ξ)D

12ξ πξ(xy

−1)∗)

∆(y)−12u(y) dξ dy,

where u is a square integrable function. The operator Op(a) is called the pseudo-differential operator with symbol a. Let

kera(x, y) = ∆(y)−12

∫G

Tr(a(x, ξ)D

12ξ πξ(xy

−1)∗)dξ.

26

Since a is in the domain of the inverse Plancherel transformation in its second variable P2,the above integral converges absolutely and

kera(x, y) = [P−12 a](x, xy−1)∆(y)−

12 .

By Plancherel’s theorem and the change of variables given by equation (1.1), we concludethat kera is a square integrable function on G × G. Hence Op(a) is a Hilbert-Schmidtoperator with kernel kera and Hilbert-Schmidt norm

‖Op(a)‖B2 = ‖kera‖L2(G×G) = ‖a‖S(G).

Now we are ready to extend the definition of Op(a) for an arbitrary symbol a ∈ S(G) usingthe previous formula and the fact that L2(G) ⊗ (B⊕1 (G) ∩ B⊕2 (G)) is a dense subspace ofS(G). Hence Op extends to a unitary map Op : S(G)→ B2(L2(G)) in a unique way.

2.2 Left and right quantizations

Having in mind the familiar Kohn-Nirenberg quantization for G = Rn (cf. (0.2) with τ = 0),one notes that for non-abelian groups there are at least two possible generalizations; a leftquantization OpL (the one used so far in this thesis)and a right quantization given by

[OpR(a)u] (x) =∫G

∫G

Tr(a(x, ξ)D

12ξ πξ(x

−1y)∗)u(y) dξ dy.

Actually, these two quantizations are related in the following sense: let a be a symbol, andconsider the symbol defined by

a(x, ξ) = πξ(x)P2P−12 a(x, ξ)πξ(x)∗.

It is an easy exercise to check that OpL(a) = OpR(a) using symbols of the form a = f ⊗ g.For unimodular groups the assignment a 7→ a is isometric, but for general non-unimodulargroups this is not the case since the modular function is not bounded.

2.3 Some operators arrising from the calculus

One of the most important families of operators in L2(G) is the one given by convolutionoperators. In this section we show how to recover the usual convolution and multiplicationoperators using pseudo-differential calculus.

27

For f, g ∈ L2(G) define the operators

[Multf u](x) = f(x)u(x),

[ConvLg u](x) =∫Gg(y)f(y−1x) dy.

In general Multf and ConvLg are not bounded operators. In fact, Multf is bounded if andonly if f is essentially bounded. Similarly, ConvLg is bounded if and only if ess sup

ξ∈G‖g(ξ)‖ <∞ nevertheless, for general non-unimodular groups the composition Multf ConvLg extendsto a Hilbert-Schmidt operator.

Suppose now that G is unimodular, and let f, g ∈ L2(G). Define the symbol a by

a(x, ξ) = f(x) g(ξ).

Using the Plancherel inversion formula one gets that the quantizations give us a very simpleway to recover these families of operators. Namely,

OpL(a) = Multf ConvLg , OpR(a)u = Multf ConvRg ,

where ConvRg is the operator given by ConvRg (u) = u ∗ g.For non-unimodular groups the picture changes dramatically, the main reason being

that the compositions Multf ConvLg are no longer Hilbert-Schmidt operators under theassumption that f, g ∈ L2(G). In fact one has that

‖Multf ConvLg ‖B2 = ‖∆−12 f‖2‖∆

12 g‖2.

For general non-unimodular groups Multf Convg is not even a bounded operator if f andg are not chosen in a suitable manner.

One way to fix this is to take functions in an appropriate dense subspace. Chosef, g ∈ L2(G) such that the functions ∆− 1

2 f,∆ 12 g are square integrable, and set

a(x, ξ) = ∆−12 (x)f(x)[∆ 1

2 g](ξ).

Then for u ∈ L2(G) we have that Op(a)u = f · (g ∗ u) for every u ∈ L2(G). Indeed,

[Op(a)u](x) =∫G

∆(x)−12 f(x)∆(xy−1)

12 g(xy−1)∆(y)−

12u(y) dy

= f(x)∫G

∆(y)−1g(xy−1)u(y) dy

= f(x)(g ∗ u)(x).

28

Another way to express the relation between symbols of the form a = f ⊗ g and operatorsof multiplication and convolution is given in the formulas

OpL(f ⊗ g) = Multf ConvLg Mult∆1/2 (2.1)

= Mult∆1/2f ConvL∆−1/2g,

OpR(f ⊗ g) = Multf ConvRg] , (2.2)

here g](x) = g(x−1). We also note that the left regular representation of G induces arepresentation acting on S(G): let a be a symbol and y ∈ G, then λy Op(a) is a Hilbert-Schmidt operator. Hence by Proposition 2.3 below there is some symbol y.a such that

λy Op(a) = Op(y.a).

It is easy to see that this defines an action of G and that

y.a(x, ξ) = πξ(y)a(y−1x, ξ).

2.3.1 Other convolution operators that appear in the literature

In [5, §1.2] the author introduces a family of convolution (to the right) operators given by

ˇConvRg u(x) =∫Gu(xy)∆(y)

12 g(y) dy = (u ∗ g∗)(x).

These operators are then used to study the space of left-invariant operators. If we want tofollow this path there are two ways to study these operators. One consists of defining

h(x) = ∆(x)−1g(x−1).

Note that h is absolutely integrable if and only if g is. Then, formally one has

OpR(f ⊗ h) = Mult∆−

12 f

ˇConvRg .

The other possibility is to put

h(x) = ∆(x)−12 g(x−1) = g∗(x).

In this case, by the definition of h and equation (2.2) one gets

OpR(f ⊗ h) = Multf ˇConvRg Mult∆−1/2 .

29

2.4 Quantization by a Weyl system

In this section we introduce the notion of a Weyl system for a general locally compactgroup. This is then used to define pesudo-differential operators through τ -quantization foran arbitrary measurable function τ : G → G. The goal in this section is to make sense offormulas (0.1) and (0.3) in a rigorous way and to clarify the role of the required hypothesis.Throughout this section we fix a Plancherel measure ν and a measurable field (πξ, Dξ)ξ∈Gas in Theorem 1.3.

We start by defining a family of integral kernels that will turn out to be very useful forthe rest of this section.

Definition 2.1. Given two square integrable functions u, v, we define the Weyl kernelassociated to u, v by

Ku,v(x, y) = ∆(y−1x)12u(y−1x)v(x) = v(x)u∗(x−1y). (2.3)

We also introduce the τ -Weyl kernels associated to the pair (u, v) as

Kτu,v(x, y) = Ku,v(τ(y−1)−1x, y), (2.4)

which just amounts to a left translation in the first variable of K.

Remark 2.1. Fubini’s theorem yields∫G

∫G|Kτ

u,v(x, y)|2 dx dy =∫G

∫G|Ku,v(x, y)|2 dx dy

=∫G

∫G|v(x)u∗(x−1y)|2 dx dy

= ‖u‖22‖v‖22.

Hence Kτu,v is square integrable. On the other hand if we identify L2-functions of two

variables with tensors of the form u ⊗ v ∈ L2(G)† ⊗ L2(G) under the identification (u ⊗v)(x, y) = u∗(y)v(x), then the adjoint of Kτ may be identified with the operator

[(Kτ )∗S](x, y) = S(τ((xy)−1)x, xy)

acting on L2(G×G). This operator is injective, hence Kτ defines a unitary isomorphism

Kτ : L2(G)† ⊗ L2(G)→ L2(G×G),

that we will denote by the same letter.

30

Remark 2.2. Note that u∗ being integrable is equivalent to ∆− 12u being integrable, moreover

‖∆− 12u‖1 = ‖u∗‖1. By equation (2.3),

∫GK

τu,v(x, y) dx = (v ∗ u∗)(y) so, if v, u∗ are

integrable, then ∫G

∫G|Kτ

u,v(x, y)| dx dy =∫G

∫G

∆(y)−12 |u(y−1)v(x)| dy dx

=∫G

∫G

∆(y)−12 |u(y)v(x)| dy dx

= ‖∆−12u‖1 ‖v‖1.

Also note that for an arbitrary y ∈ G, by Hölder’s inequality∫G|Kτ

u,v(x, y)| dx ≤ ‖∆12u‖2‖v‖2.

By the previous remarks we see that if u, v belong to an appropriate dense subspace,like Cc(G) for example, then it makes sense to take the Plancherel transform of Kτ

u,v inboth its first and second variables.

Definition 2.2. Let τ : G→ G be given a measurable function, x ∈ G and a representationπξ in the class of ξ ∈ G. For nice enough Θ ∈ L2(G;Hξ), in a sense we specify below,define the τ-Weyl System by

[W τ (πξ, y)Θ](x) = ∆(y−1x)12πξ(τ(y−1)x)D

12ξ (Θ(y−1x)).

We drop the index τ in the notation when the choice τ(·) = e is made. It is important tonote that these operators are only defined for elements in

Θ ∈ L2(G;Hξ) | Θ(x) ∈ Dom(D12ξ ) µ-a.e. and ∆

12D

12ξ Θ ∈ L2(G;Hξ).

The domain of W τ (πξ, y) contains the space of vectors of the form η ⊗ u, where η ∈Dom(D

12ξ ) and u ∈ Dom(Mult∆1/2). Here Multf denotes the operator of multiplication by

f , so in particular W τ (πξ, y) is a densely defined operator.

Proposition 2.1. Let ν be a fixed Plancherel measure and let (πξ, D12ξ )

ξ∈G be a measurablefield as in Theorem 1.3. The operators (W τ (πξ, y))(ξ,x)∈G×G form a measurable field ofdensely defined closed operators on

∫⊕GHξ dξ⊗L2(G). If πξ and π′ξ are unitarily equivalent

representations with intertwining operator U , then

W τ (πξ ′, y) = (U ⊗ IdHξ)W τ (πξ, y) (U∗ ⊗ IdHξ).

Because of this, once we fix a measurable field of representations and Duflo-Moore

31

operators, we will just write W τ (ξ, y) instead of W τ (πξ, y).

Proof. Let (ηξ)ξ∈G be a measurable section of∫⊕GHξ dξ such that ηξ ∈ Dom(D

12ξ ) for ν-

almost everywhere, and let u be a square integrable function such that ∆1/2u is also squareintegrable, then

W τ (πξ, y)(ηξ ⊗ u) = πξ(τ(y−1) · )D12ξ ηξ ⊗ [λy Mult

∆12u], (2.5)

is clearly a measurable section of Hξ ⊗ L2(G) ∼= L2(G;Hξ). In particular ηξ ⊗ u ∈Dom(W τ (πξ, y)). Let (ηiξ)ξ∈Gi∈N be a total subset of

∫⊕GHξ dξ such that ν-a.e. each

ηiξ is in the domain of D12ξ , and let ujj∈N ⊆ Cc(G) be a total subset of L2(G). Then

(ηiξ ⊗ uj)ξ∈Gi,j∈N is a total subset of

∫⊕GHξ dξ ⊗ L2(G) contained in the domain of

(W τ (πξ, y))(ξ,x)∈G×G; and the map

(ξ, y) 7→ 〈W τ (πξ, y)(ηiξ ⊗ uj), (ηi′ξ ⊗ uj

′)〉

is measurable for each i, j, i′, j′ ∈ N. Hence the Weyl system forms a measurable field ofdensely defined operators. By equation (2.5), one sees that it is a composition of closedoperators, hence closed for each pair (ξ, y) ∈ G×G. Let U be a unitary operator such that

πξ′(x) = Uπξ(x)U∗

for each x ∈ G. Let D′ξ = UDξU∗. By the semi-invariance relation and equation (1.12),

this is the Duflo-Moore operator associated to π′ξ. Then one has

W τ (π′ξ, y)(ηξ ⊗ u) = [Uπξ(τ(y−1) · )U∗D′ξ12 ηξ]⊗ [λy Mult

∆12u]

= [Uπξ(τ(y−1) · )D12ξ U∗ηξ]⊗ [λy Mult

∆12u]

= (U ⊗ IdHξ)W τ (πξ, y)(U∗ηξ ⊗ u)

and therefore the required relation

W τ (πξ ′, x) = (U ⊗ IdHξ)W τ (πξ, x) (U∗ ⊗ IdHξ).

Given two square integrable functions u, v such that ∆ 12u ∈ L2(G), and a given vector

32

φ ∈ Dom(D1/2ξ ), define the operator Wτ

u,v : L2(G)→ L2(G) by

Wτu,v(ξ, y)φ =

∫G

(W τ (ξ, y) u⊗ φ) (x)v(x) dx.

We defined this operator on a dense subspace, but it can be extended to a Hilbert-Schmidtoperator (cf. Proposition 2.2 below). Note that

Wτu,v(ξ, y)φ = πξ(τ(y−1))

∫G

(∆

12 u)

(y−1x) v(x)πξ(x)D12ξ φdx

= πξ(τ(y−1))P1(Ku,v)(ξ, y)φ

=(P1K

τu,v

)(ξ, y)φ.

Here P1 denotes the Plancherel transform in the first variable. This short remark leads tothe following proposition.

Proposition 2.2. The assignment u⊗ v 7→ Wτu,v extends to a unique unitary map Wτ :

L2(G)† ⊗ L2(G)→ S(G) called the Fourier-Wigner τ-transformation.

Proof. Given two square integrable functions u, v such that ∆1/2u is also square integrable,applying Plancherel theorem one gets

‖Wτu,v‖2S(G) =

∫G

∫G‖Wτ

u,v(ξ, y)‖2B2 dξ dy

=∫G

∫G‖(P1K

τu,v)(ξ, y)‖2B2 dξ dy

=∫G

∫G|Kτ

u,v(x, y)|2 dx dy

= ‖u‖22‖v‖22,

which shows that W extends to a unitary isomorphism.

We introduce the Wigner τ-transformation of two functions u, v ∈ L2(G) as

Vτu,v = P2P−11 W

τu,v = P2K

τu,v ∈ S(G).

More explicitly,

Vτu,v(x, ξ) =∫G

∆(y−1τ(y−1)−1x)12u(y−1τ(y−1)−1x)v(τ(y−1)−1x)πξ(y)D

12ξ dy.

We record for further use the orthogonality relations, valid for u, u′, v, v′ ∈ L2(G)

⟨Wu,v,Wu′,v′

⟩S(G) = 〈v, v′〉〈u′, u〉 =

⟨Vu,v,Vu′,v′

⟩S(G) . (2.6)

33

2.5 Pseudo-differential operators

As before, fix a measurable map τ : G → G. In the next definition we formalize theτ -quantization Opτ (a) introduced in equation (0.1). When τ is the constant functionτ(x) = e we drop the superscript in the notation.

Definition 2.3. Let a ∈ S(G) be a symbol with Plancherel transform in both variablesa = P1P−1

2 a ∈ S(G). Define Opτ (a) to be the unique bounded linear operator in L2(G)defined by the relation

〈Opτ (a)u, v〉 = 〈a,Wτu,v〉S(G).

Or equivalently,〈Opτ (a)u, v〉 = 〈a,Vτu,v〉S(G).

Opτ (a) is then called the τ-pseudo-differential operator with symbol a, while the mapa 7→ Opτ (a) will be called the τ-pseudo-differential calculus or τ-quantization.

Note that

|〈Opτ (a)u, v〉| ≤ ‖a‖S(G)‖Wu,v‖S(G) = ‖a‖S(G)‖u‖2‖v‖2.

So in particular ‖Opτ (a)‖ ≤ ‖a‖S(G). By Theorem 1.3, a different choice of Plancherelmeasure and tuple (πξ, Dξ)ξ∈G gives rise to an equivalent calculus.

We have the following proposition. With only small its proof follows that of [33,Theorem 3.8].

Proposition 2.3. Let us define

Λu,v(w) = 〈w, u〉v, ∀ w ∈ L2(G),

the rank-one operator associated to the pair (u, v). Then one has

Λu,v = Opτ (Vτu,v), ∀ u, v ∈ L2(G).

In particular, the mapping Opτ sends S(G) unitarily onto the Hilbert space of all Hilbert-Schmidt operators in L2(G).

Proof. Note that relation (2.6) gives

〈Opτ (Vτu,v)u′, v′〉 = 〈Vτu,v,Vτu′,v′〉

= 〈v, v′〉〈u′, u〉

= 〈Λu,v(u′), v′〉,

34

for all square integrable functions. Hence Λu,v = Opτ (Vτu,v). Since the rank-one operatorsare dense in the space of Hilbert-Schmidt operators, the desired conclusion holds.

After working out the formulas and assuming that u, v belong to an appropriate densesubset, and that for µ-almost all x ∈ G the operator a(x, ξ) satisfies the hypothesis ofProposition 1.5, one has

〈Opτ (a)u, v〉 = 〈P−12 a,Kτ

u,v〉L2(G×G)

=∫G

∫G

∫G

Tr(a(x, ξ)D

12ξ πξ(y)∗

)Kτu,v(x, y) dξ dx dy.

Thus by making the substitution y 7→ yx and using (2.4)

[Opτ (a)u](x) =∫G

∫G

Tr(a(τ(y−1)x, ξ)D

12ξ πξ(y)∗

)∆(y−1x)

12u(y−1x) dξ dy

=∫G

∫G

Tr(a(τ((xy)−1)x, ξ)D

12ξ πξ(xy)∗

)∆(y)−

12u(y−1) dξ dy

=∫G

∫G

Tr(a(τ(yx−1)x, ξ)D

12ξ πξ(yx

−1))

∆(y)−12u(y) dξ dy.

Thus, the kernel of Opτ (a) is the square integrable function

kerτa(x, y) = ∆(y)−12

∫G

Tr(a(τ(yx−1)x, ξ)D

12ξ πξ(xy

−1)∗)dξ (2.7)

= ∆(y)−12 [P−1

2 a](τ(yx−1)x, xy−1). (2.8)

Formula (2.8) shows in another way that Opτ is unitary. Indeed by consecutive use ofFubini’s theorem,∫

G

∫G|kerτa(x, y)|2 dx dy =

∫G

∫G

∆(y)−1∣∣∣P−1

2 a(τ(xy−1)x, xy−1)∣∣∣2 dx dy

=∫G

∫G

∣∣∣P−12 a(τ(y)x, y)

∣∣∣2 dx dy= ‖a‖2S(G).

We summarize the most important properties of Opτ in the following theorem (cf.Section 2.6 for the definition of an H∗-algebra).

Theorem 2.1. The τ -quantization Opτ : S(G)→ B2(L2(G)

)is a unitary isomorphism of

Hilbert spaces. Additionally Opτ has the following properties

(i) If τ(x) = e, then Opτ (f ⊗ g) = Multf Convg Mult∆1/2.

35

(ii) The integral kernel of Opτ (a) is given by

kera(x, y) = ∆(y)−12 [P−1

2 a](τ(yx−1)x, xy−1).

Remark 2.3. Suppose G is unimodular. If τ(x) = e for all x ∈ G, then formula (2.7) reads

kera(x, y) =∫Gd

1/2ξ Tr

(a(x, ξ)πξ(xy−1)∗

)dξ = [P−1

2 a](x, xy−1).

Remark 2.4. When G = Rn, under the identification of G with Rn defined by ξ(x) =e−2πi〈ξ,x〉, with Haar measure as the Plancherel measure, and setting Dξ = IdL2(R), werecover the Kohn-Nirenberg calculus (cf. eq. (0.2)).

2.5.1 Relations between different τ-quantizations

The choice of measurable function τ : G → G has to do with ordering issues in thequantization arising from the non-commutativity of the operators involved. One may askfor is the exact relation between the quantizations given by Op and Opτ . Let a ∈ S(G) bea symbol defined on the group, and consider the unitary map Ωτ : L2(G×G)→ L2(G×G)given by

ΩτS(x, y) = S(τ(y)−1x, y).

Then if we consider the symbol

aτ = P2 (Ωτ )∗ P−12 a,

after successive uses of the changes of variables shown in equation (1.1) we arrive at therelation between the quantizations

Op(aτ ) = Opτ (a).

This relation shows how to pass from the quantization where τ(x) = e to an arbitraryτ -quantization. Note that a 7→ aτ is a unitary isomorphism.

Example 2.1. Consider τ(x) = x. Then if a = f ⊗ g we have

Opτ (a) = ConvLg Multf Mult∆1/2 .

With formula (2.1) in mind one sees that this choice of τ changes the order in which theoperators of multiplication and convolution are composed.

36

Example 2.2. We implement a right τ -quantization via the formula

[OpτR(a)u](x) =∫G

∫G

Tr(a(τ(yx−1)x, ξ)D

12ξ πξ(x

−1y)∗)u(y) dξ dy.

As in the previous example, if we take τ(x) = x the operator with symbol a = f ⊗ g is

OpτR(a) = ConvRg] Multf .

2.6 Involutive algebras of symbols

The fact that Opτ is an isomorphisms allows us to define a product ∗τ , which we will callthe Moyal product, and an involution ∗τ on S(G) by the formula

Opτ (a ∗τ b) = Opτ (a) Opτ (b),

Opτ (a∗τ ) = Opτ (a)∗.

With this extra structure S(G) becomes anH∗-algebra [7, Appendix A], i.e. a completeHilbert algebra, which we will denote by H∗(G). Being an H∗-algebra means that thefollowing relations hold,

〈a ∗τ b, c〉H∗(G) = 〈b, a∗τ ∗τ c〉H∗(G),

〈a, b〉H∗(G) = 〈b∗τ , a∗τ 〉H∗(G),

for all a, b, c ∈ H∗(G). These relations follow from Proposition 2.3 and the fact that theHilbert-Schmidt operators are a H∗-algebra with the usual composition and involution.

37

Chapter 3

Some concrete examples ofnon-unimodular groups

In this chapter we work out in detail the representation theory of the affine group andof Grélaud’s group, two simple examples of non-unimodular groups. Then we show howquantization works in theses examples.

3.1 A pseudo-differential calculus for the affine group

In this section we develop a pseudo-differential calculus on the affine group of the realline. The theory of unitary representations of the affine group is worked out in [17] or in[12, §6.7]. In this section

G = (b, a) ∈ R2 | a 6= 0,

denotes the Affine group, with product law

(b, a) · (b′, a′) = (ab′ + b, aa′).

The group G is a Lie group and the connected component of the identity is a simplyconnected Lie group; it is also the semi-direct product RoR×, where R× = R \ 0 is themultiplicative group of R acting on R by multiplication. Since it is also a real algebraicgroup, it is type I.

Let g = R2 be the Lie algebra of G with bracket defined by

[(β, α), (β′, α′)] = (αβ′ − α′β, 0).

The left Haar measure is |a|−2 da db, and its right Haar measure is given by |a|−1da db.

38

Hence the modular function is∆(b, a) = |a|−1.

The Haar measures of G are a product of a continuous function on G and the Lebesguemeasure of R2. Hence they are strongly equivalent measures. The connected componentof the affine group is the only connected simply connected (non-unimodular) Lie group ofdimension 2 with Lie algebra g.

One important fact is that the exponential map exp : g → G is a diffeomorphism of gonto the connected component of the identity. The exponential and logarithm maps aregiven by

exp(β, α) =(β

α(eα − 1), eα

)log(b, a) =

(b

a− 1 log(a), log(a)),

with the limit case being used if a = 1.

3.1.1 Representation theory of the Affine group

One of the special properties of G is that its unitary dual consists of a point with positivePlancherel measure equal to 1 and a ν-null set of one-dimensional representations (cf.[12, §6.7] where the Mackey machine is used to compute the unitary dual).

Up to a set of zero measure, G consist of a single representation π called the quasi-regular representation. It acts on Hξ = L2(R) via

π(b, a)f(x) = |a|1/2e−2πixbf(ax).

We denote the equivalence class of π in G by the Greek letter ξ. This is a square-integrableirreducible representation since it has positive Plancherel measure.

The Duflo-Moore operator corresponding to the representation ξ is given, on its naturaldomain, by

Dξf(x) = |x|f(x).

An explicit calculation of the matrix coefficient functions for f, g ∈ L2(R) shows that

Cf,g(b, a) = |a|1/2∫Rf(x)g(ax)e2πibx dx.

39

Let ua(x) = f(x)g(ax). Then

‖Cf,g‖22 =∫G|ua(−b)|2

db da

|a|

=∫G|f(b)g(ab)|2 db da

|a|

= ‖f‖22‖D− 1

2π g‖22,

This is another way of seeing that π is a square-integrable representation. The Planchereltransform is given by

[f(ξ)g](x) = |x|1/2∫Gf(b, a)g(ax)e2πibx db da

|a|

= |x|1/2∫G

1|a|F1f(x, a)g(ax) da

=∫G

|x|12

|a|F1f

(x,a

x

)g(a) da.

This is a Hilbert-Schmidt operator with integral kernel

Kf (x, a) = |x|12

|a|F1f

(x,a

x

), (3.1)

where F1 denotes the usual Fourier transform on the real line acting in the first variable.A calculation in [12, §6.7] shows that the Plancherel formula implies

‖Kf‖2L2(R2) =∫∫

R2

|x||a|2

∣∣∣∣F1f

(x,a

x

)∣∣∣∣2 dx da=∫∫

R2|F1f(x, a)|2 dx da

|a|2

= ‖f‖2L2(G).

So, as required by the general theory, the Plancherel transform implements a unitaryisomorphism

P : L2(G)→ B2(L2(R)).

Note that the center of the right-hand side von Neumann algebra is C · IdL2(G), so thePlancherel transform implements a central decomposition of L2(G). In particular, thiscalculation shows that the support of the Plancherel measure is ξ. There is not enoughroom in L2(G) for another representation, hence we identify the unitary dual of G withthe singleton ξ.

40

3.1.2 The quantization for the affine group

Let τ(b, a) = (0, 1) for all (b, a) ∈ G. For a symbol α formula (0.1) reads

[Op(α)u](a, b) =∫∫

R2

1|a′|3/2

Tr(α(b, a)D

12ξ π

(b− a

a′b′,

a

a′

)∗)u(b′, a′) db′ da′.

If we identify symbols on G with functions f ∈ L2(G×G) via the unitary map defined byequation (3.1) we may think of symbols as integral operators of the form

[α(x, ξ)u](y) =∫RKf(x,·)(y, s)u(s) ds = [P2f(x, ξ)u](y).

Using formula (2.8) for kera, formula (0.1) boils down to

[Op(α)u](b, a) =∫∫

R2

1|a′|3/2

f

((b, a),

(b− a

a′b′,

a

a′

))u(b′, a′) db′ da′.

In particular the map

L2(G×G) 3 f 7→ kerP2f ∈ B2(L2(G)),

is a unitary equivalence.

3.1.3 Operators that arise from the representation

Let A = (0, 1) and B = (1, 0) be the generators of g; they satisfy the commutation relation[A,B] = B. Then if dπ(X)u = d

dtπ(etX)u |t=0, denotes the (densely defined) inducedrepresentation of g on the Hilbert space L2(R) we have that

[dπ(A)f ](x) = 12f(x) + xf ′(x),

[dπ(B)f ](x) = 2πixf(x).

Clearly [dπ(A), dπ(B)] = dπ(B). Note also that

dπ(A) = 12

(x · d

dx+ d

dx· x),

is the infinitesimal generator of dilations of R, a well-studied operator on R.

41

3.2 Grélaud’s group

Grélaud’s group is one of the non-unimodular Lie groups that arises from Bianchi’s classi-fication of 3-dimensional Lie algebras [3]. Let θ ∈ R \ 0 and let

A = Aθ =(

1 θ

−θ 1

).

We endow Gθ = R× R2 with the multiplication law

(s, u) · (s′, u′) = (s+ s′, e−s′Au+ u′).

Here

etA = et(

cos(tθ) sin(tθ)− sin(tθ) cos(tθ)

).

Gθ is a semi-direct product Ro R2 with unit e = (0, 0) and inverse given by

(s, u)−1 = (−s,−esAu).

The left Haar measure coincides with the Lebesgue measure on R3, but the group is notunimodular. Indeed, the modular function is given by

∆(s, u) = e−2s.

The Lie algebra gθ of Gθ is, as a vector space, R× R2 with the Lie bracket

[(σ, µ), (σ′, µ′)] = (0, σAµ′ − σ′Aµ).

Since the commutator [gθ, gθ] is contained in 0×R2, a commutative subalgebra, then gθ

is a two-step solvable Lie algebra. Its exponential map exp : gθ → Gθ is given by

exp(σ, µ) =(σ,− 1

σA−1

(e−σA − Id

)µ

).

The exponential map is clearly a diffeomorphim. In particular Gθ is an exponentiallysolvable Lie group, connected and simply connected as a topological space. Recall thatexponentially solvable groups are type I. Most of the representation theory of Grélaud’sgroup is worked out in detail in [13, §4.4], but we calculate it in Section 3.2.2 to show howTheorem 1.1 works.

42

3.2.1 Representation theory of Gθ

There are two families of representations of Gθ. Let λ ∈ R, then

χλ(s, u) = eiλs

is a one-dimensional unitary representation. Let p ∈ S1 be a unit vector in the plane.Then we have the unitary representation on Hp = L2(R)

πp(s, u)f(t) = e−i〈p,e(s−t)Au〉f(t− s).

The following proposition is proven in [13, §4.4]. We also give a proof of this propositionin Section 3.2.2 using Proposition 1.1.

Proposition 3.1. Every irreducible infinite-dimensional unitary representation of Gθ isunitarily equivalent to πp for some p ∈ S1. Moreover the set of classes of these representa-tions is a ν-conull set on Gθ, and the Plancherel measure coincides with a multiple of theHaar measure on the circle.

From now on we identify the Plancherel measure of Gθ with the Haar measure on theunit circle having total measure ν(S1) = 1. We also identify πp with p ∈ S1. The map F2

denotes the usual Fourier transform on R2 on the second variable.The Duflo-Moore operators, applied to a function f lying on a dense subset of L2(R),

are given byDpf(t) = 1

2πe−2tf(t).

With this in mind, we find that the Plancherel transformation on Gθ is given by

[f(p)g](t) = 1√2π

∫∫R×R2

f(t− s, u)e−i〈e−sAtp,u〉e−sg(s) ds du

= 1√2π

∫RF2f

(t− s, 1

2πe−sAt

p

)e−sg(s) ds,

where At denotes the transpose of the matrix A. Let p(ϕ) = (cos(ϕ), sin(ϕ)). Then

43

Plancherel’s formula reads∫S1‖f(p)‖2B2 dp = 1

2π

∫S1

∫R

∫R

∣∣∣∣F2f

(t− s, 1

2πe−sAt

p

)∣∣∣∣2 e−2s dt ds dp

= 14π2

∫ 2π

0

∫R

∫R

∣∣∣∣F2f

(t,

12πe

−sAtp(ϕ)

)∣∣∣∣2 e−2s dt ds dϕ

=∫∫

R×R2|F2f(t, u)|2 dt du

= ‖f‖22,

where the second equality comes from the fact that the Jacobian of the transformation

u(s, ϕ) = 12πe

−sAt(

cos(ϕ)sin(ϕ)

),

is e−2s/4π2, i.e. e−2sds dϕ = 4π2du.

3.2.2 Computing the unitary dual

Here we compute the unitary dual of G = Gθ using the Mackey machine (Theorem 1.1).Let N = 0×R2 and let H = G/N ∼= R. Note that N is a type I normal closed subgroupof Gθ since

(s, u) · (0, v) · (s, u)−1 = (0, esAv).

We identify an element µ ∈ R2 with a representation σµ ∈ N via

σµ(0, u) = e−i〈µ,u〉.

After the above identification, the dual action of G on R2 is given by

(s, u).µ = esAtµ.

Since ‖esAtµ‖ = es‖µ‖, one can see that the orbits are spirals towards the origin, and the

origin itself. Hence N/H may be identified with the unit circle S1 and a point 0, whichis a countably separated Borel space. Let µ ∈ S1. Note that the stabilizer in G of µ is N .Let πµ = IndGN (σµ) be the induced representation acting on the space L2(G,N, σµ, µR),where µR is the Lebesgue measure on R. Thus, f ∈ L2(G,N, σµ, µR) satisfies

f((s, u) · (0, v)) = σµ(0, v)∗f(x) a.e and∫R|f(s, 0)|2 ds <∞.

44

Note that the Hilbert space on which πµ acts is unitarily isomorphic to L2(R) with thefollowing unitary isomorphism: identify a function f ∈ L2(R) with the function f0 ∈L2(G,N, σµ, µR) given by

f0(s, u) = ei〈µ,u〉f(s).

Abusing notation, πµ is given on L2(R) by

πµ(s, u)f(t) = e−i〈µ,e(s−t)Au〉f(t− s).

Having in mind that the modular function coincides with the multiplication by ∆, one alsohas that the Duflo-Moore operators are, up to normalization, given by

Dµf(t) = e−2tf(t).

If we start with the orbit of µ = 0, then the stabilizer of this point is all of the group,and the trivial representation extends to the whole group. Hence the representations weare looking at are of the form χλ = IndGG(1 × σλ), where σλ((s, u)N) = e−iλs is a unitaryrepresentation of Hµ

∼= R for some λ ∈ R. It is easy to see that IndGG(σλ) is unitarilyequivalent to the one-dimensional representation

χλ(s, u) = e−iλs.

Now we see thatGθ = πµ | µ ∈ S1 ∪ χλ | λ ∈ R ∼= S1 ∪ R.

To use Proposition 1.4 we chose Ω = R2 \ 0 as our G-invariant νN -conull subset of N ,then we have the following ν-conull subset of G

⋃Oσ∈Ω/H

IndGN (σ × ρ) | ρ ∈ 0 = πµ | µ ∈ S1.

Since for each σ ∈ Ω, the dual of the stabilizer Hσ consist of only a point, the calculationof the Plancherel measure amounts to find a pesudo-image of the Plancherel measure ofN ∼= R2 on Ω/H ∼= S1 (i.e. the Lebesgue measure on R2). The Haar measure of the circleis a fine choice of pseudo-image. Hence the Plancherel measure of Gθ is just the Haarmeasure on the circle.

45

3.2.3 The quantization

Let a ∈ S(Gθ) be a symbol and let τ : G → G be the map defined by τ(s, u) = (0, 0) forall the elements of the group. Then, the kernel of the pseudo-differential operator withsymbol a is given by

kera(u, s, u′, s′) =∫S1

Tr(a(s, u, p)D

12p πp

(s− s′, es′Au− e−s′Au′

))e2s′dp.

An the symbol space can be identified with

S(G) = L2(G)⊗∫ ⊕GB2(L2(R)) dξ

∼= L2(G)⊗ L2(G)⊗ B2(L2(R))∼= L2(G× S1)⊗ B2(L2(R))∼= L2(R5 × [0, 1]),

the last space endowed with the Lebesgue measure.

46

Chapter 4

Crossed products of C∗-algebras

We introduce some tools from the theory of crossed products of C∗-algebras, this in turnshall help us to cover the bigger class of compact operators on L2(G), using the Schrödingerrepresentation of a natural crossed product associated to G, namely C0(G) oG.

4.1 C∗-dynamical systems

Definition 4.1. A C∗-dynamical system is a triplet (A, G, α), where G is a locally com-pact group, A is a C∗-algebra and α : G→ Aut(A) is a strongly continuous representationof G.

Let (A, G, α) be a C∗-dynamical system, to it which we associate the space L1(G;A)of all Bochner-integrable functions f : G → A. This space has the structure of a Banach∗-algebra with convolution and involution laws given by formulas

(f ? g)(x) =∫Gf(y)αy

(g(y−1x)

)dy,

f?(x) = ∆(x)−1αx(f(x−1)∗

).

The Banach ∗-algebra L1(G;A) is naturally isomorphic to the projective tensor productA⊗ L1(G). Consider the universal norm on L1(G;A) given by

‖f‖AoG = supρ‖ρ(f)‖,

where the supremum is taken over the set of all non-degenerate ∗-representations of A. Thecrossed product AoG is the enveloping C∗-algebra of L1(G;A), that is, its completionunder the norm ‖·‖AoG.

Example 4.1. Let A be a C∗-algebra, take G to be the trivial group and α to be the

47

trivial representation, then AoG is naturally isomorphic to A. Another more interestingexample is when we have a continuous action of G on a topological space X. This induces amap α : G→ C0(X) given by αx(f)(p) = f(x−1p). Then (C0(X), G, α) is a C∗-dynamicalsystem and it encapsulates all the information of the group action.

Definition 4.2. A covariant representation of a C∗-dynamical system (A, G, α) iscomposed of: a unitary representation π of G and a non-degenerate ∗-representation ρ ofA, both acting on a Hilbert space H in such a way that they satisfy the relation

π(x)ρ(f)π(x)∗ = ρ(αxf) f ∈ A, x ∈ G.

We denote this data as the triplet (ρ, π,H).

Example 4.2. Let (C0(X), G, α) be the C∗-dynamical system induced by an action of Gon X. Then a covariant representations of C0(X) oG is exactly the same as a system ofimprimitivity (cf. [31] §3.7). In fact there is a one-to-one correspondence between contin-uous actions of a group G and C∗-dynamical systems (A, G, α) where the C∗-algebra Ais an abelian one. This can be easily seen from the fact that every abelian C∗-algebra isof the form C0(X) for some locally compact space and there is a correspondence betweenstrongly continuous representations α : G → Aut(C0(X)) and continuous actions of G onX [44, Proposition 2.7]. In particular, a system of imprimitivity is the same as a covariantrepresentation of a C∗-dynamical system where the C∗-algebra is abelian.

Every covariant representation (ρ, π,H) of a C∗-dynamical system naturally induces anon-degenerate ∗-representation ρ o π of the crossed product A o G on H, which is theunique extension of the representation of L1(G;A) given by the integral

ρo π(f) =∫Gρ (f(y))π(y) dy. (4.1)

This process sets up a bijection between the covariant representations of a C∗-dynamicalsystem and the non-degenerate ∗-representations of the crossed product associated to it [44,Proposition 2.40].

4.2 The Schrödinger representation

There is a natural covariant representation associated to any left-invariant C∗-algebraof functions defined on G. We show some of its properties and how it relates to ourquantization.

Let A be a left-invariant C∗-subalgebra of the space of bounded left uniformly contin-uous functions on G. For an A-valued function F on G and elements x, z ∈ G we make

48

the convenient identificationF (x)(z) = F (z, x).

The triplet (A, G, α) is a C∗-dynamical system when endowed with the action α : G →Aut(A) given by αxF (z, y) = F (z, x−1y). Then our convolution and involution laws aregiven by

(F ? G)(z, x) =∫GF (z, y)G(y−1z, y−1x) dy,

F ?(z, x) = ∆(x)−1F (x−1z, x−1).

Let H denote the space of square integrable functions on G. Then we have a naturalcovariant representation of the triplet (A, G, α) given by

λxu(y) = u(x−1y), Multf u(y) = f(y)u(y).

The Schrödinger representation is the integrated representation Sch = Multoλ ofAoG. More explicitly for a function F ∈ L1(G;A),

[Sch(F )u](x) =∫GF (x, y)u(y−1x) dy

=∫GF (x, xy−1)∆(y)−1u(y) dy.

The good thing about the Schrödinger representation is that, formally, one gets the fol-lowing relation

Op(f ⊗ g) = Sch(f ⊗ g) Mult∆1/2 .

We can estimate the norm

〈Sch(f)(u), v〉 ≤∫G

∣∣∣∣∫Gf(x, y)u(y−1x)v(x) dy

∣∣∣∣ dx≤∫G

∣∣∣∣∫G‖f(y)‖∞u(y−1x)v(x) dy

∣∣∣∣ dx≤∫G|(‖f‖∞ ∗ u)(x)v(x)| dx

≤ ‖f‖L1(G;A)‖u‖2‖v‖2.

But much more is true, from the integrated form formula (4.1) one sees that in fact onehas a better estimate of this norm:

‖Sch(f)‖ ≤ ‖f‖AoG.

49

4.2.1 Interlude on Amenable groups

Let π1, π2 be unitary representations ofG. Each one lifts to a non-degenerate ∗-representationπ1 and π2 of the C∗-algebra C∗(G) of the group. We say that π1 is weakly contained inπ2 if and only if kerπ2 ⊆ kerπ1 .

A locally compact group G is called amenable if the trivial representation is weaklycontained in the left regular representation.

We recall some of the equivalent definitions for amenability [7].

Proposition 4.1. The following conditions are all equivalent.

(i) The group G is amenable.

(ii) There is a bounded linear functional L∞(G)→ R that is positive and left-invariant.

(iii) All of the irreducible representations of G are weakly contained in the left regularrepresentation.

(iv) The support of any Plancherel measure is all of G.

In general a connected Lie group is amenable if and only if it has a closed normal solvablesubgroup such that the quotient is compact.

By Proposition 4.1 non-compact semisimple Lie groups are not amenable.

Example 4.3. Some examples of amenable groups are: finite groups, abelian groups andconnected solvable Lie groups. An extension of an amenable group by another amenablegroup is also amenable. Every quotient by a closed normal subgroup and every closedsubgroup of an amenable group is amenable.

Example 4.4. Since Grélaud’s group is a connected solvable Lie group, it is amenable.Similarly the affine group is an extension of R by R×, hence it is also amenable.

Remark 4.1. In [44, §4.4] it is shown that there is epimorphism of C∗-algebras betweenthe space of compact operators in L2(G) into C0(G) o G. Part of the result is that thismorphism is an isomorphism if and only if the group is amenable. In particular, for eachf ∈ C0(G) o G, the operator Sch(f) is a compact operator with operator norm equal tothe universal norm of f . This gives us a way to extend our quantization so that we coverthe more general case of compact operators. In the general case that G is not amenable,we still have that Sch : C0(G) oG→ B0(L2(G)) is an onto contraction.

50

Chapter 5

Conclusions

There is still much room for more general quantization. For example, one could dropthe type I hypothesis since the non-unimodular Plancherel theorem still works partiallyin this setting [8]. Another, more important aspect to improve is the generality of thesymbols involved, and to get an analogue of the Hörmander symbol classes, for at leastconnected simply connected Lie groups. This has already been done for compact andnilpotent connected Lie groups [10, 36]. This is particularly important since almost allthe operators that appear in mathematics and physics are unbounded, and our class onlycovers the much smaller class of Hilbert-Schmidt operators. And the usual Kohn-Nirenbergcovers a big class of unbounded operators which are needed for applications.

51

Alphabetical Index

C∗-dynamical system, 47H∗-algebra, 37τ -Weyl system, 31τ -pseudo-differential calculus, 34τ -quantization, 34

affine group, 38amenable, 50

Borel space, 7

contragradient representation, 10countably separated Borel space, 7countably separated measure, 7covariant representation, 48crossed product of C∗-algebras, 47

direct integral of Hilbert spaces, 8dual action, 17Duflo-Moore operator, 20, 23

Fourier algebra, 24Fourier-Wigner τ -transformation, 33

group algebra, 14Grélaud’s group, 42

induced representation, 17intertwining operator, 10invariant measure, 16irreducible representation, 10

Mackey Borel strucure, 14matrix coefficients, 19measurable field of operators, 9measurable field of Hilbert spaces, 8

measurable field of representations, 13measurable section, 8measurable transversal, 7modular function, 12Moyal product, 37multiplicity-free representation, 11

non-unimodular Plancherel theorem, 21

operator-valued Fourier transform, 20

Plancherel measure, 15Plancherel transform, 21primary representation, 11projective representation, 10pseudo-differential operator, 26, 34pseudo-image measure, 7

quansi-regular representation of theaffine group, 39

quasi-equivalent representations, 10quasi-invariant measure, 16quotient Borel structure, 7

regularly embedded, 18

Schrödinger representation, 49square integrable representation, 20standard Borel space, 7standard measure, 7strongly continuous representation, 9strongly equivalent, 7strongly quasi-invariant measure, 16subrepresentation, 10

trivial Mackey obstruction, 18

52

two-sided regular representation, 13type I group, 11type I representation, 11

unimodular group, 12unitarily equivalente representations, 10unitary dual, 14

unitary representation, 9universal norm, 14, 47

weakly containment of representations,50

Weyl kernel, 30Wigner τ -transformation, 33

53

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