Dirac induction for semisimple Lie groups

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MSc Mathematical Physics

Master Thesis

Dirac induction for semisimple Lie groups

Author: Supervisor:

Didier Collard

dr. H.B. Posthuma

Examination date:

February 18, 2016

Korteweg-de Vries Institute for Mathematics

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2

Abstract

Let G be a connected Lie group. We construct the reduced group C∗-algebra C_r∗(G)as the closure of the group algebra L1(G)in the regular representation, and consider its K-theory. Dirac induction gives a ho-momorphism:

µ : R(K)→ K_∗(C_r∗(G)) [V ]7→ index DV.

where K is a maximal compact subgroup of G and R(K) is the free abelian group on classses of irreducible representations of K. It sends each irreducible representation to the index of the twisted Dirac operator DV.

In this thesis a precise construction of the Dirac induction homomorphism is given, and it is calculated explicitly forR2_{, compact Lie groups and SL(2,}_{R). We then define the higher order index as a generalisation}

of the L2-index. This index is defined using cyclic cocycles obtained from the Lie algebra cohomology and the pairing between K0(Cr∗(G))and cyclic cohomology. Using the higher order pairing, we describe

K0

(

C_r∗(R2))and discuss how to generalise this to SL(2,R) and general semisimple Lie groups.

Title: Dirac induction for semisimple Lie groups

Author: Didier Collard, didier.collard@student.uva.nl, 10022147 Supervisor: dr. H.B. Posthuma

Second Examiner: dr. R. Bocklandt Examination date: February 18, 2016 Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 105-107, 1098 XG Amsterdam http://kdvi.uva.nl

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Introduction

Let G be a group. If G is finite, it is possible to define the group algebra.

Definition (Group algebra). The group algebra of a finite group G is given by the functions:

C[G] = {f : G → C} , with as product:

(f ∗ g) (x) =∑

u∈G

f (u)g(u−1x).

It is a basic result from representation theory that the representations of G are the same as the representa-tions ofC[G], and Maschkes theorem gives a decomposition of C[G] in terms of its representations. Now if

Gis a locally compact group, using the Haar measure we can also define its group algebra.

Definition (Reduced group C∗ algebra). The reduced group C∗ algebra of a locally compact group G is given by:

C_r∗(G) ={f : G → C : compactly supported},

where· denotes the completion of Cc(G)in the left regular representation on the space of square integrable

functions L2_{(G). The product is given by the convolution product:}

(f ∗ g) (x) = ∫

f (y)g(y−1x)dy.

If G is abelian the Fourier transform gives a description of the group C∗algebra, and if G is a compact Lie group, the Peter-Weyl theorem gives a decomposition Cr∗(G) =

⊕

φ∈ ˆKHφ⊗ H∗φ. For a general connected

Lie group, we want to describe the K-theory of the group C∗algebra. It is possible to do this using the Dirac

induction homomorphism:

µ : R(K)→ K_∗(C_r∗(G)) [V ]7→ index DV,

where K is a maximal compact subgroup of G and R(K) is the free abelian group on the classses of ir-reducible representations of K. It maps each irir-reducible representation to the index of the twisted Dirac operator DV. This Dirac operator DV gives a Fredholm operator on a Hilbert C∗-module over Cr∗(G), for

which an index with values in K0(Cr∗(G))is defined.

It was conjectured independently by Connes and Kasparov in 1982 that µ gives an isomorphism of groups. In 1987 Wasserman gave a proof of this conjecture for connected reductive Lie groups, proving it in partic-ular for semisimple Lie groups. However, as we will discuss, this proof does not give much insight into the connection between R(K) and K_∗(C_r∗(G))by Dirac induction.

Analysing the Dirac induction map for semisimple Lie groups is the main goal of this project. In the first three chapters, we give a precise construction of the Dirac induction map. In the last chapter, we try to analyse the Dirac induction map using higher order index pairings.

Prerequisites for Dirac induction

First, a precise construction of the group C∗algebra is required. Let G be a locally compact group. In order to construct its group algebra, we need to establish the correspondence between group representations and

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Introduction 5

L1(G)representations. If φ : G → U (Hφ)is a unitary group representation, it induces a L1(G)

represen-tationφ : Le 1_(G)_{→ B (H} φ)by: e φ(f ) = ∫ f (x)φ(x)dx, f ∈ L1(G).

We show that this assignment gives an equivalence of categories between G and L1_(G)_{representations. As}

L1(G)is not a C∗ algebra, we show that by completing in the operator norm onB(L2(G))using the left regular representation, we obtain the reduced group C∗algebra C_r∗(G).

Next, we need to construct the Dirac operator. We first describe the general construction of a Dirac op-erator, which is a first order elliptic differential operator acting on sections of a vector bundleV over a Riemannian manifold X. We show that if X has a spin structure, it is possible to define a spinor bundleS over X, and then define the Dirac operator using the Levi-Civita connection and the Clifford action onS . This leads to the following result:

Proposition. Let X be a spin manifold of dimension n. Then the spinor representation Spin(n)→ S gives a

spinor bundleS of Clifford modules over Cl(X) and the Spin-invariant metric together with the Levi-Civita connection makesS into a Dirac bundle. It thus has a Dirac operator:

D : Γ(S ) → Γ(S ).

Finally, it is shown that it is possible to twist a spinor bundle with another Hermitian vectorV over X. This gives the twisted Dirac operator acting onS ⊗ V . This twisted Dirac operator construction is used in the Dirac induction map, as each K-representation V gives a vector bundle over G/K.

Construction of the Dirac induction homomorphism

Let G be a Lie group. Then G has a maximal compact subgroup K, and using the Haar measure we can give

Gan inner product with is both left and right invariant under the action of K. This inner product induces a metric on G/K, and makes G/K into a Riemannian homogeneous G-space.

Using the adjoint representation Ad_K we construct a spin structure on G/K. This allows to define the spinorS bundles on G/K. As each K-representation V gives a vector bundle G ×KV over G/K by the

associated vector bundle construction, we can twist the spinor bundle be taking the tensor product withV . This is again a Dirac bundle, thus for every K-representation V we obtain a twisted Dirac operator DV.

Finally, a Cr∗(G)-module structure on the sections Γ (V ⊗ S ) is constructed. The Dirac operator gives a

Fredholm operator acting on Γ (V ⊗ S ), therefore we can take the index and obtain an element index DV ∈

K0(Cr∗(G)). This gives the Dirac induction homomorphism:

µ : [V ]7→ index DV ∈ K0(Cr∗(G)) .

Connes-Kasparov conjecture & Baum-Connes conjecture

Around 1982, independently by Kasparov [Kasparov, 1983, §7] and Connes [Baum and Connes, 1982, §5] the following conjecture was formulated.

Conjecture (Connes-Kasparov conjecture (1982)). Let j = dim(G/K) mod 2. Dirac induction gives an

isomorphism of groups:

µ : R(K)→ Kj(Cr∗(G))

V 7→ index DV.

The other K-group is trivial, i.e. Kj+1Cr∗(G) = 0.

This conjecture is a special case of the Baum-Connes conjecture. This was also formulated around 1982, the present form is stated in [Baum et al., 1994]:

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6 Introduction

Conjecture (Baum-Connes conjecture). Let K_jG(EG) be the equivariant K-homology groups with G-compact support. A class in KG

j (EG), is represented by a G-equivariant elliptic operator on EG, where

EGis the universal example for proper actions of G. Then the assembly map

µ : K_jG(EG)→ Kj(Cr∗(G)) ,

which assigns to each operator its index is an isomorphism.

It is possible to identify the representation ring R(K) with K_jG(G/K), where j = dim G/K mod 2.

G/K is the universal example EG for proper actions of G, and in this way the Dirac induction map be-comes the Baum-Connes assembly map [Baum et al., 1994, prop. 4.12]. A recent discussion of the relation between the Baum-Connes conjecture, the Connes-Kasparov conjecture and the discrete series can be found in [Lafforgue, 2002].

The Baum-Connes conjecture is true for many important classes of groups, see [Chabert et al., 2003, §1] and [Lück and Reich, 2005, §2.6.3]. However for the more general Baum-Connes conjecture with coefficients, counter-examples have been constructed [Higson et al., 2002]. In the case of semisimple Lie groups, the proof is provided by [Wassermann, 1987].

The proof of Wasserman uses an calculation of Cr∗(G)using the Plancherel decomposition of the regular

representation, see remark 3.4.14. The idea of the Baum-Connes conjecture is to get insight into Cr∗(G), not

to use insight in C_r∗(G)to prove something about Dirac induction. As the proof of Wasserman only uses a trick to calculate C_r∗(G)directly, it gives not much insight into the relation between R(K) and K_∗(C∗r(G)).

Goal of this research project

The goal of this project is to analyse the Dirac induction map for a semisimple Lie group using index pairings. Dirac induction gives for every representation a G-equivariant elliptic differential operator. In [Atiyah and Schmid, 1977], the L2_{index of such an operator is defined, and an index theorem is given. More generally,}

this index can be viewed as a pairing between cyclic cohomology and K0(Cr∗(G))using the Connes-Chern

character. However, one pairing is not enough to describe the complete K-theory, therefore we define (see [Pflaum et al., 2015b, §2.2]) the following:

Definition. Let c be a cocycle in HC2k(AG)such that c extends to suitable intermediate algebraAGwith

AG⊂ AG ⊂ Cr∗(G). Then the higher order index pairing induced by c is given by:

⟨Ch2k(·), c⟩ : K0(Cr∗(G))→ C,

where⟨·, ·⟩ is the pairing between HCeven(AG)and K0(Cr∗(G))from the Connes-Chern character.

We discuss a method to obtain these cyclic cocycles from the Lie algebra cohomoloy H_Lie∗ (g, K). These higher index pairings can then be used to describe elements in K0(Cr∗(G)), and thus analyse the Dirac

induction map.

In this thesis, we precisely construct the Dirac induction map, and then compute it explicitly for G =R2

and compact Lie groups. These give ‘toy models’ for the more difficult case SL(2,R). We have been able to describe K0

(

C_r∗(R2))using this new higher order index pairing. Also, we have formulated ideas on how to generalise this to SL(2,R) and general semisimple Lie groups, but we were not able to calculate the higher order index paring in these cases yet.

Structure of this thesis

This thesis is structured as follows.

1. Group C*-algebras: We begin with constructing the group C∗algebra for a general locally compact group. Then we compute the group algebra for abelian and compact Lie groups.

2. Spin geometry & Dirac operators: The goal of this chapter is to discuss the required constructions from Spin geometry to construct the Dirac operator.

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Introduction 7

3. Dirac induction: In this chapter, we define the Dirac induction homomorphism. We finish by computing it for G =R2_{, compact Lie groups and G = SL(2,}_R)

4. Higher index theory: In this chapter, we define the higher order index, and show how to obtain index pairings for cycles in the Lie algebra cohomology. We then compute the higher order index pairing for

G =R2, and show that the higher index is required to describe K0

(

C_r∗(R2)). We finish with a discus-sion of how the generalise this to SL(2,R) and general semisimple Lie groups.

Finally in the appendices, the following is discussed.

A. Hilbert C∗-modules: We define Hilbert C∗-modules and Fredholm operators, and discuss the required results that we need to compute the index of a Fredholm operator.

B. Differential geometry: We discuss principal bundles and the associated vector bundle construction, as this is very important in Dirac induction. Also, the definition of (pseudo)-differential operators on a manifold are given.

Acknowledgements

First of all, I would like to thank Hessel Posthuma for supervising my thesis. Our cooperation during the process of writing my thesis has been flexible, yet very productive. When problems occurred, I could always stop by his office, and during our meetings he always took the time to explain everything thoroughly.

Particularly in the last two months, he spent a lot of time to help me finish my project. We made substantial progress during the last weeks: we constructed the ‘volume index’ and solved all the theoretical obstacles to calculate it for SL(2,R). Our discussions about the higher index were very inspiring, and I regret not being able to study this further. Furthermore, I am very grateful to Hessel for helping me finish my masters degree in mathematics next to my medicine study.

I would also like to thank Raf Bocklandt for being the second examiner for my thesis. Finally, I would like to express my gratitude to all the fellow students in the master room for the breaks, lunches, discussions, and their support. Without them, obtaining my masters would have been almost impossible.

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1. Group C

∗

algebras

The goal of this chapter is to construct the group C∗ algebra of a locally compact group. First, the group algebra of a locally compact will be defined, then the relation between representations of the group and representations of the group algebra will be discussed. Using the representations of the group, it is possible to define a new norm on the group algebra and complete it in this norm to a C∗ algebra. At the end the Peter-Weyl theorem for compact groups and the Fourier transform for abelian groups are treated, and using these results the C∗algebra of compact and abelian groups is computed.

1.1. Group algebra of a locally compact group

In this section, the group algebra L1_(G)_{of a locally compact group G will be defined. We follow [Folland,}

1995, ch. 2], the proofs omitted in this section can be found there.

Haar measure

Let G be a locally compact topological group, and denote by Lx, Rx left and right translation with x∈ G.

We recall the existence and some important properties of the Haar measure.

1.1.1 Definition. A left (respectively right) Haar measure on G is a non-zero Radon measure µ on G that

satisfies µ (xE) = µ(E) (respectively µ(Ex) = µ(E)) for every Borel set E⊂ G and every x ∈ G. This is equivalent with requiring:

∫

f ◦ Lydµ =

∫

f dµ, ∀f ∈ Cc(G)and y ∈ G,

where for the right Haar measure one should read Ry instead of Ly.

1.1.2 Theorem. Every locally compact group G possesses a left Haar measure. This measure is unique up

to a positive constant.

Proof. If G is a Lie group, one can construct the Haar measure by creating a G-invariant volume form using

left translations, see [Duistermaat and Kolk, 2000, §3.13]. If G is only a topological group, the Haar measure is constructed using a linear functional on Cc(G)and the Riesz representation theorem, see [Folland, 1995,

thm. 2.10].

In the rest of the chapter we fix a Haar measure on G, and normalize∫ 1 dµ = 1if G is compact. Define the Lpspaces for 1≤ p ≤ ∞ by:

1.1.3 Definition. The space Lp_(G)_{for 1}_{≤ p < ∞ is given by:}

Lp(G) = { f : G→ C : ∥f∥p = (∫ |f|p_dµ )1/p <∞ } ,

and L∞(G)is given by:

L∞(G) ={f : G → C : ∥f∥_∞=ess sup|f| < ∞} .

The next result allows us to analyse the Lp_{spaces using the dense subset of compactly supported}

contin-uous functions.

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1.1. Group algebra of a locally compact group 9

1.1.4 Theorem ([Rudin, 1974, thm. 3.14]).

(i) The compactly supported functions Cc(G)lie dense in Lp(G)for each 1≤ p < ∞.

(ii) The compactly supported functions Cc(G)lie dense in C0(G)in the∥ · ∥∞norm.

The modular function relates the left Haar measure to the right Haar measure.

1.1.5 Proposition. There exists a unique continuous homomorphism ∆ : G→ R>0such that:

∫

f◦ Rydµ = ∆

(

y−1) ∫ f dµ, for all f ∈ Cc(G).

∆is called the modular function of G. G is called unimodular if ∆ = 1, which implies that µ is also right invariant.

1.1.6 Remark. If µ is a left invariant Haar measure on G, then ∆(y−1)µ(y)is a right invariant Haar measure on G. It thus relates the left to the right Haar measure on G.

1.1.7 Proposition ([Folland, 1995, prop. 2.29]). The following groups are unimodular:

(i) If G is compact, then G is unimodular. (ii) If G/[G, G] is compact, then G is unimodular.

(iii) If G is connected and a semisimple Lie Group, then [g, g] = g, hence G = [G, G], so G is unimodular. The following lemma is useful for computations with the Haar measure:

1.1.8 Lemma. Let λ be a left Haar measure on G and ρ a right Haar measure on G. Then the following

substitution rules hold:

dλ(x−1)= ∆(x−1)dλ(x), dρ(x−1)= ∆(x)dρ(x).

Convolution product

In order to make L1_(G)_{into a Banach-* algebra, a product and involution on L}1_(G)_{must be defined. First}

the convolution product on L1_(G)_{is defined and some convergence properties are derived. Then an}

invo-lution is defined, and it is shown that together with the convoinvo-lution product it makes L1(G)into a Banach-* algebra.

1.1.9 Definition. Let f∈ L1(G), g∈ Lp(G). Define the convolution of f and g by: (f∗ g)(x) =

∫

f (y)g(y−1x)dy. (1.1)

The Minkowski’s Inequality for Integrals is required to show convergence of the convolution product.

1.1.10 Lemma (Minkowski’s Inequality for Integrals, [Folland, 1999, thm. 6.19]). Suppose that (X,M, µ)

and (Y,N , ν) are σ-finite measure spaces, and let f be an M ⊗ N -measurable function on X × Y . (i) If f ≥ 0 and 1 ≤ p < ∞, then:

[∫ (∫ f (x, y)dν(y) ]p dµ(x) )1/p ≤ ∫ [∫ f (x, y)pdµ(x) ]1/p dν(y).

(ii) If 1≤ p ≤ ∞, f(·, y) ∈ Lp_(µ)_{for almost every y and the function y}_{7→ ∥f (·, y) ∥}

pis in L1(ν), then:

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10 1. Group C∗algebras

• the function x7→∫f (x, y)dv(y)∈ Lp(µ)and ∫ f (·, y)dν(y) p ≤ ∫ ∥f(·, y)∥pdν(y).

Using this inequality, we can prove the next proposition which shows convergence of the convolution product and gives a bound on its norm.

1.1.11 Proposition. Let 1≤ p ≤ ∞, f ∈ L1_(G)_{and g} _{∈ L}p_{(G). Then:}

(i) The integral in (1.1) converges absolutely for almost every x and f∗ g is in Lp(G). (ii) The norm is bounded by∥f ∗ g∥p≤ ∥f∥1∥g∥p.

(iii) If p =∞, then f ∗ g is continuous.

(iv) If G is unimodular, (i-iii) are true for g∗ f. If f is compactly supported, (i) and (iii) are true for g ∗ f.

Proof. Let f∈ L1_(G)_{and g} _{∈ L}p_{(G). Using Minkowski’s inequality, we obtain:}

∥f ∗ g∥p =

∫ f (y)(g◦ L_y−1dy)

p

≤∫ g ◦ Ly−1 _p|f(y)|dy = ∥f∥1∥g∥p.

This proves (i) and (ii).

Now for (iii), note that we can rewrite the convolution product as:

f∗ g(x) = ∫ f (xy)g(y−1)dy. For x, x0∈ G we get: |f ∗ g(x) − f ∗ g (x0)| ≤ ∥g∥∞ ∫

|f(xy) − f (x0y)| dy. (1.2)

If we let f ∈ Cc(G), then f is uniformly continuous, hence the|f(xy) − f (x0y)| term in (1.2) is small

when|x − x0| is small. Using that f has compact support and that Ryis continuous, it follows that f∗ g is

continuous. Now if hn→ h in L1(G), then hn∗ g → h ∗ g in the ∞-norm. As Cc(G)is dense in L1, this

shows that f∗ g is continuous for all f ∈ L1_(G)_{and g}_{∈ L}∞_(G).

For (iv), let f ∈ L1_(G)_{and g}_{∈ L}p_(G)_{and rewrite the convolution product as:}

∥g ∗ f∥ = ∫ (g◦ R_y−1)f (y)∆(y−1)dy

p

.

We proceed in the same manner, and apply Minkowski’s inequality to obtain:

≤∫ (g ◦ Ry−1) _p|f(y)|∆ ( y−1)dy =∥g∥p ∫ support f |f(y)|∆(y)(1/p)−1_dy_≤ ( sup supportf ∆(y)(1/p)−1 ) ∥g∥p∥f∥1, (1.3)

which proves (i) if f has compact support, and (i-ii) if G is unimodular. To show (iii) for g ∗ f, let f ∈

Cc(G), g∈ L∞(G)and use the following expression for the convolution product:

g∗ f(x) =

∫

g(xy−1)f (y)∆(y−1)dy.

Using this formula one can show in the same manner as for f ∗ g that g ∗ f is continuous. Finally if G is unimodular, it follows from (1.3) that if hn → h in L1(G)the product g∗ hnconverges to g∗ h in the

∞-norm. Using denseness of Cc(G), this shows that g∗ h is continuous if G is unimodular, h ∈ L1(G)and

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1.2. Group and group algebra representations 11

If G is discrete, the Dirac delta function at the identity δegives the unit for the convolution product. If G

is not discrete, such function does not exist, but we can use an approximate identity which approximates the delta function.

1.1.12 Proposition. LetZ be a neighbourhood base at e in G. For each Z ∈ Z let ηZ be a symmetric

indicator function around Z, i.e. a compactly supported function such that ηZ ≥ 0, ηZ

(

x−1)= ηZ(x)and

∫

ηZ = 1. Then:

(i) ∥f ∗ ηZ− f∥p → 0 as Z → {e}, if 1 ≤ p < ∞ and f ∈ Lp, or if p = ∞ and f is right uniformly

continuous.

(ii) ∥ηZ∗ f − f∥p → 0 as Z → {e}, if 1 ≤ p < ∞ and f ∈ Lp, or if p = ∞ and f is left uniformly

continuous.

1.1.13 Remark. DefineZ as the collection of open sets containing {e}. If we order them by inclusion, Z

becomes a net. We can define limZ→{e} as the limit over the sets inZ, and the proposition above implies

that for any f ∈ L1_(G)_{it holds that:}

lim

Z→{e}f ∗ ηU = f, Ulim→{e}ηU∗ f = f.

In this manner we obtain an approximate identity for the algebra L1_(G).

Now define the following involution on the group algebra L1(G):

1.1.14 Definition. Define an involution for f ∈ L1_(G)_by,

f∗(x) = f (x−1)∆(x−1).

It is easy to check that that∗ defines an involution. It follows from proposition 1.1.11 that ∥f ∗ g∥L1 ≤

∥f∥L1∥g∥_L1for all f, g∈ L1(G), hence L1(G)is a Banach∗-algebra. To summarize this section:

1.1.15 Proposition. L1_(G)_{with product and involution for f, g}_{∈ L}1_(G)_{defined by:}

(f ∗ g)(x) = ∫

f (y)g(y−1x)dy f∗(x) = f (x−1)∆(x−1),

is a Banach∗-algebra.

1.2. Group and group algebra representations

In this section the definitions of group and algebra representations are given and the correspondence between them is discussed. Using the representations of L1_{(G), two new norms are defined on L}1_{(G). By taking the}

closure of L1_(G)_{in these norms we obtain the group and the reduced group C}∗_{algebra. We follow [Folland,}

1995, §1.5, ch. 3] and [Davidson, 1996, ch. VII].

Relation between group and group algebra representations

Let G be a locally compact group and L1_(G)_{its group algebra. We begin by defining a group algebra and a}

unitary group representation.

1.2.1 Definition. Let A be a Banach∗-algebra. A ∗-representation of A on a Hilbert space H is a

∗-homo-morphism φ : A→ B (H). We call φ non-degenerate if there is no non-zero v ∈ H such that φ(x)v = 0 for all x∈ A.

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1.2.2 Definition. A unitary representation is a group homomorphism φ : G → U (Hφ), where U (Hφ)is

the group of unitary operators on some non-zero Hilbert spaceHφ, which is continuous with respect to the

strong operator topology, i.e. the map x7→ φ(x)u is continuous for every u ∈ Hφ.

Two important examples of unitary representations are the left and right regular representation of G on

L2(G).

1.2.3 Example.

(i) The left regular representation φLof G on L2(G)is the representation given by:

[φL(y)f ] (x) = f

(

y−1x), f ∈ L2(G).

(ii) The right regular representation φRof G on L2(G)is the representation given by:

[φR(y)f ] (x) = ∆(y)1/2f (xy) , f ∈ L2(G).

We can now discuss the relation between unitary representations of the group G and∗-representations of the group algebra L1_{(G). Let φ : G} _{→ U (H}

φ) be a unitary representation of G. Then it induces a

non-degenerate representationφ : Le 1(G)→ B (Hφ)by:

e

φ(f ) =

∫

f (x)φ(x)dx, f ∈ L1(G). (1.4)

Here the operated valued integral is defined as the operator satisfying:

⟨ eφ(f )u, v⟩ =

∫

f (x)⟨φ(x)u, v⟩ dx, f ∈ L1(G), u, v∈ Hφ.

1.2.4 Proposition. Let φ : G → U (Hφ) be a unitary representation of G. For each f ∈ L1(G),φ(f )e

defined by (1.4) is a bounded linear operator with∥ eφ(f )∥ ≤ ∥f∥_L1. This gives a non-degenerate L1(G)

representation:φ : Le 1_(G)_{→ B(H}

φ).

Proof. Let f, g∈ L1(G), u, v∈ Hφ. As φ is unitary:

⟨φ(f)u, v⟩ ≤ ∥f∥L1∥u∥∥v∥,

henceφ(f )e is a bounded linear operator with∥ eφ(f )∥ ≤ ∥f∥L1. We continue with checking thatφedefines

an algebra homomorphism:

⟨ eφ (f∗ g) u, v⟩ =∫ ∫ ⟨f (y)g(y−1x)φ(x)u, v⟩dydx =

∫ ∫

⟨f(y)g (x) φ(yx)u, v⟩ dydx

= ⟨∫ ∫ f (y)g(x)φ(y)φ(x)u, v ⟩ dxdy =⟨φ(f)φ(g)u, v⟩ , ⟨ eφ (f∗) u, v⟩ = ∫ ⟨ ∆(x−1)f (x−1)φ(x)u, v ⟩ dx = ∫ ⟨ f (x)φ(x−1)u, v ⟩ dx = ∫ ⟨u, f(x)φ (x) v⟩ dx = ⟨( eφ (f ))∗u, v⟩

Furthermore it is clear thatφ(λf + g) = λe φ(f ) +e φ(g)e for λ∈ C, hence eφis a∗-representation. Finally to show that it is non-degenerate let v ̸= 0 ∈ Hφ. Using an approximate unit, it follows that there exists a

compact neighbourhood V of e∈ G such that:

|φ(x)u − u| ≤ ∥u∥, ∀x ∈ V.

Set f = _{|V |}1 χV, where χ is the indicator function on V . Then:

∥ eφ(f )u− u∥ = 1 |V |

∫_V [φ(x)u− u] dx < ∥u∥,

where in the last step we used the intermediate value theorem for integrals. We conclude thatφ(f )ue ̸= 0,

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Let eψ : L1(G)→ B(Hψ)be a group algebra representation. To go back and construct a group

represen-tation fromφe, the algebra representation needs to be evaluated at a function which peaks at a specific group element, so we want to define:

ψ(g) = eψ (δg) .

In order to define this formally, we use translates of the approximate identity to approximate the delta func-tion at each group element.

LetZ be a neighbourhood base at e, and denote by (ηZ)the approximate identity from proposition 1.1.12.

Now let eψ : L1(G)→ B(Hψ)be a representation of L1(G)which is non-degenerate. Then define a unitary

representation ψ : G→ U (Hψ)of G by:

ψ(x) = lim

Z→{e}

e

ψ (ηZ◦ Lx−1) , x∈ G. (1.5)

where Lx−1is left multiplication by x−1.

1.2.5 Proposition. Let eψ : L1(G)→ B(Hψ)be a representation of L1(G)which is non-degenerate. Then

ψdefined by (1.5) is a unitary representation of G. This representation is independent of the choice of µ.

Proof. We start with defining ψ on the finite span:

D = span{ψ(f )v, ve ∈ Hψ

}

.

As eψis non-degenerate, D⊥ = 0, henceD lies dense in Hφ. Now let τx be left translation with x−1 for

x∈ G, so τx(y) = f ( x−1y). Note that: lim Z→{e} e ψ ((τxηZ)∗ f) v = ψ (τxf ) v,

hence the operators eψ (τxηZ)converge strongly to an operator on ψ on D with:

ψ(x) eψ(f )v = eψ (τxf ) v. (1.6)

Because eψis a∗-representation, it has operator norm 1, hence we have |ψ(x)∥ ≤ ∥τxηZ∥ = 1 for all x ∈ G.

This shows that ψ(x) is a bounded operator, hence it extends uniquely to a bounded linear operator onHψ.

It follows from (1.6) that ψ defines a representation onD, hence its extension is also a representation. Note that for all x∈ G and v ∈ Hψ:

∥v∥ = ψ(x−1)ψ (x) v ≤ ∥ψ(x)v∥ ≤ ∥v∥,

hence ψ(x) is an isometry, so it defines an unitary representation.

Finally, we have to show that ψ is continuous. Let x∈ X and (xα)a sequence in G with xα → x. Note

that τxαf → τxffor all f ∈ L

1_{(G). This implies that ψ (x}

α)→ ψ(x) strongly on D. Now let v ∈ Hφ, and

(vn)a sequence inD with vn→ v. An ϵ/3 argument using:

∥ψ (xα)∥ ≤ ∥ψ (xα) u− ψ (xα) un∥ + ∥ψ (xα) un− ψ (x) un∥ + ∥ψ (x) u − ψ(x)un∥ ,

shows that ψ is also continuous on Hψin the strong topology. We conclude that ψ defines an unitary

repre-sentation L1_(G)_{→ U (H}

ψ). Equation (1.6) shows that ψ is independent of the choice of µ.

The unitary representations of G and the non-degenerate algebra representations both form categories with intertwining operators as morphisms. Denote these by respectively GRep and ARep. The assignments defined in (1.4) and (1.5) between group and algebra representations can be viewed as functors between GRep and ARep by sending each intertwining operator to the same intertwining operator. These functors give an equivalence of categories.

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1.2.6 Theorem. The functors defined in (1.4) and (1.5) gives an equivalence of categories between the

cate-gory GRep(G) of unitary representations of G and the catecate-gory ARep(L1_(G))_{of non-degenerate algebra}

representations of L1_(G).

Proof. Denote the functor GRep(G)→ ARep(L1(G))withF and the functor ARep(L1(G))→ GRep(G) withG. Let eψ : L1_(G)_{→ B (H}

ψ)be a group algebra representation, let φ =G( eψ)be the induced

represen-tation of G andφ =e F(φ) the L1(G)-representation coming from φ. Now let f, g∈ L1_{(G), then:}

e ψ(f ) eψ(g) = eψ(f∗ g) = ∫ f (y) eψ (τyg) dy = [∫ f (y)φ(y)dy ] e ψ(g) =φ(f ) ee ψ(g).

This shows that eψ =φeon the dense subsetD, hence they are also equal on Hψ. This shows thatF ◦ G = id.

Next, we show thatF is injective. Suppose that π : G → U (Hψ)is another unitary representation such

thateπ = eψ. Then for all u, v∈ Hψand f ∈ L1(G)we have

∫ f (x)⟨π(x)u, v⟩ dx = ⟨eπ(f)u, v⟩ = ⟨ ] ψ(x)u, v ⟩ = ⟨ ] φ(x)u, v ⟩ = ∫ f (x)⟨φ(x)u, v⟩ dx, hence:

⟨π(x)u, v⟩ = ⟨φ(x)u, v⟩ , ∀u, v ∈ Hψ,

so π(x) = ψ(x) for all x ∈ G, so F is also injective. This shows that F and G give a bijection between unitary group representations and non-degenerate group algebra representations.

Finally, if φ, ψ are unitary representations of G it follows directly from the definition that an operator

Hφ → Hψintertwines φ, ψ if and only if it intertwines eψ,φe, so we conclude thatF, G give an equivalence

of categories between GRep(G) and ARep(L1(G)).

From now on, we will denote both the group and the algebra representation by φ, and omit the tilde. An important example is given by the representation of L1_(G)_{induced by the left regular representation}

of G. This representation will be used in the next next section to construct a C∗-norm on L1_(G).

1.2.7 Example. The left regular representation φL: G→ U

(

L2(G))induces the following representation of L1_(G):

φL: L1(G)→ B

(

L2(G))

f 7→ φL(f )with φL(f )g = f∗ g for g ∈ L2(G).

This representation is faithful, to see this we use an approximate identity. LetZ be a neighbourhood base at e, and denote by (ηZ)the approximate identity from proposition 1.1.12. Now let f ∈ L1(G)with φL(f ) = 0.

As ηZ ∈ L2(G)for each Z, we have f∗ ηZ = 0for all Z ∈ Z. But as

∥f ∗ ηZ− f∥L1

Z→{e}

−→ 0,

it follows that∥f∥_L1 = 0, hence f = 0.

Construction of group C∗algebra

Let G be a locally compact group. In the previous section it has been shown that L1(G)is a Banach∗-algebra. It is however not a C∗algebra, because the∗-identity does not hold, as the next example shows.

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1.2.8 Example. Let G = (R, +) and define f ∈ Cc(G)by:

f (x) =      1 + x if x∈ [−1, 0] 1− x if x ∈ [0, 1] 0 else . Then f∗ = f,∥f∥2_L1 = 1and ∥f∗f∥_L1 = ∫ 0 −1(1 + x) 2_{dx +} ∫ 1 0 (1− x)2= 2 3, hence the∗-identity ∥f∗f∥_L1 =∥f∥2_L1 does not hold.

There are two ways to make L1_(G)_{into a C}∗_{algebra. In both constructions, the operator norm of}

repre-sentations of L1_(G)_{is used. For the group C}∗_{algebra, we take the supremum over the operator norm of all}

representations of L1(G). This is well defined as for any representation φ of G it holds that:

∥φ(f)∥_B(Hφ)≤ ∥f∥1, f ∈ L

1_(G),

by proposition 1.2.4. As for any f ∈ L1_{(G), we have:}

sup φ {∥φ (a ∗_a)_{∥} = sup} φ { ∥φ (a)∥2}₌ ( sup φ {∥φ (a)∥} )2 ,

it follows that sup{∥φ(f)∥ : φ is a representation of G} defines a C∗norm.

For the reduced group C∗algebra, only the norm of the left regular representation is used. As both norms are C∗norms, we can the closure with respect to these norms, to obtain the group algebra and the reduced group C∗algebra.

1.2.9 Definition.

(i) The group C∗algebra C(G) of G is the completion of L1(G)in the norm:

∥f∥C∗(G)=sup{∥φ(f)∥ : φ is a representation of G} .

(ii) The left regular representation gives an injection: φL : L1(G) ,→ B

(

L2(G)). Define the reduced

group C∗algebra as the closure of L1(G)inB(L2(G)):

C_r∗(G) := φL(L1(G)).

1.2.10 Remark. We have now constructed the (reduced) group C∗algebra by completing L1_(G)_{with a C}∗

norm. It is also possible to begin with the compactly supported functions Cc(G). From proposition 1.2.4 it

follows that:

∥φL(f )∥_B(L2_(G))≤ ∥f∥_C∗_(G)≤ ∥f∥₁, f ∈ L1(G).

Using this equation and denseness of Cc(G)in L1(G), it follows that Cc(G)is also dense in Cr∗(G)and

C∗(G). We will use this later when in the construction of modules over Cr∗(G)starting from Cc(G).

Any∗-representation of L1_(G)_{extends to a representation of C}∗_{(G). In particular, the left regular}

repre-sentation extends to a homomorphism:

φL: C∗(G)→ Cr∗(G). (1.7)

We have the inequality∥φL(f )∥_B(L2_(G)) ≤ ∥f∥_C∗_(G)for all f ∈ L1(G)hence any convergent sequence in

φL(C∗(G))comes from a convergent sequence in C∗(G). This implies that φLis a surjective

homomor-phism onto φL(C∗(G)) = Cr∗(G).

This map is not injective in general. Groups for which C_r∗(G) = C∗(G)are described by amenability. For comprehensiveness, the main results about amenability are cited, but as we are only using the reduced group

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1.2.11 Definition. A group G is called amenable if there is a left invariant translation mean for G. This

is a positive linear functional m : L∞(G) → C of norm 1 with m (f ◦ Lx) = m (f )for all x ∈ G and

f ∈ L∞(G).

1.2.12 Theorem. Let G be a locally compact group. The left regular representation φL: C∗(G)→ Cr∗(G)

is an isomorphism if and only if G is amenable.

1.2.13 Remark. The following groups are amenable:

(i) Compact groups: the left translation invariant mean is constructed by averaging using the Haar mea-sure.

(ii) Abelian groups: Using the Fourier transform, one can show that φLis an isometry, hence bijective, see

proposition 1.4.7. It is also possible to construct an left translation invariant mean directly using the fixed point theorem from Karkov and Katukani, see [Davidson, 1996, thm. VII.2.1].

1.2.14 Remark. We will be using the reduced group C∗-algebra, as the complete C∗algebra is to big for a general semisimple Lie group. For example, for SL(2,R), the reduced group C∗algebra can be constructed from the discrete and principal series of SL(2,R), see 3.10. The corresponding classes in K-theory can be reached by Dirac induction, but if we would use all unitary representations of SL(2,R), the algebra would be too big.

1.3. Compact Lie groups

Let K be a compact Lie group. The goal of this section is to determine the group C∗algebra of K. In order to do this, the representation theory of compact groups is discussed. Schur orthogonality and the Peter-Weyl theorem are stated, which give a decomposition of the left regular representation that is used in the construction of C_r∗(G). We follow [Folland, 1995, ch. 5] for this. Using these results, the group C∗algebra

C_r∗(K)and its K-theory are determined, using the method from [Landsman, 1998, thm. 1.8.1].

1.3.1 Remark. In this chapter we use inner products which are linear in the first variable. The inner product

on L2_(K)_{is thus given by:}

⟨f, g⟩ =

∫

K

f (x)g∗(x)dx.

For the decomposition of the regular representation, we need to determine all the irreducible representa-tions of K. The next results show that they are all finite dimensional and unitary.

1.3.2 Theorem. Let K be a compact group. Then every irreducible representation of K is finite dimensional,

and every unitary representation of K is a direct sum of irreducible representations.

1.3.3 Remark (Weyl’s Trick). Note that as K is compact, any finite dimensional representation can be

equipped with an inner product for which the representation is unitary by averaging using the Haar measure. Let (·, ·) be any inner product on a representation Hφ, then:

⟨v, w⟩ :=

∫

K

(φ(x)v, φ(x)w) dx,

is an inner product whcih makes φ unitary. One can therefore see every irreducible representation of K as a unitary representation.

For each unitary representation φ of K, we define the matrix elements of a representation. These matrix elements give the matrix coefficients of φ with respect to an orthonormal basis ofHφ.

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1.3. Compact Lie groups 17

1.3.4 Definition. Let φ : K → U (Hφ)be a unitary representation of K. Let u, v∈ Hφ. A matrix element

of φ is a function of the form:

φu,v(x) :=⟨φ(x)u, v⟩ .

Denote the span of all matrix elements of φ by:

Eφ =span{⟨φ(x)v, u⟩ : u, v ∈ Hφ} ⊂ C(G).

IfHφis finite dimensional, define dφ := dimHφ. Then define for an orthogonal basis

{

e1, . . . , edφ

} the matrix elements φij by:

φij(x) := φej,ei =⟨φ(x)ej, ei⟩ .

Denote by ˆKthe set of equivalence classes of finite dimensional continuous irreducible representations of

K. The Schur orthogonality relations describe the matrix elements for each equivalence class in ˆK.

1.3.5 Theorem (Schur Orthogonality Relations). Let φ, ψ be irreducible unitary representations of K,

de-note by [φ], [ψ] their class in ˆKand considerEφ,Eψas subspaces of L2(K). Then:

(i) If [φ]̸= [ψ] then Eφ ⊥ Eψ.

(ii) If{e1, . . . edφ

}

is an orthonormal basis forHφ, then:

{√

dφφij : i, j = 1, . . . , dφ

}

.

is an orthonormal basis forEφ.

1.3.6 Remark. The same is also true for a general locally compact group G, if we take continuous finite

dimensional unitary irreducible representations.

From Schur orthogonality and separability of L2_{(K), it follows that there are only countable many}

irre-ducible representations of K.

1.3.7 Proposition ([Duistermaat and Kolk, 2000, thm. 4.3.4]). Let G be a locally compact group. If the

topology of G has a countable basis, then ˆGis countable. In particular, if K is a compact group, then ˆKis countable.

The matrix coefficients also define elements in L1(G). The following proposition reformulates the Schur orthogonality relations in terms of the convolution product on L1(G).

1.3.8 Proposition ([Dixmier, 1977, thm. 14.3.3, 14.3.7]). Let φ, ψ be inequivalent irreducible unitary

repre-sentations of K.

(i) If u1, v1 ∈ Hφand u2, v2∈ Hψ, then:

φu1,v1 ∗ φu2,v2 = 0. (ii) If u1, v1, u2, v2 ∈ Hφthen ⟨φu1,v1, φu2,v2⟩ = ∫ K φu1,v1(x)φu2,v2(x)dx = 1 dφ ⟨u 1, u2⟩ ⟨v2, v1⟩ , φu1,v1∗ φu2,v2 = 1 dφ ⟨v2 , u1⟩ φu2,v1. (iii) In particular, if{e1, . . . edφ }

is an orthonormal basis forHφ, then:

φi,j∗ φk,l =

1

dφ

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We can now state the Peter-Weyl theorem, which gives a decomposition of the left regular representation

φL: K → L2(K).

1.3.9 Theorem (Peter-Weyl theorem). Let K be a compact group and denote by ˆKthe set of equivalence classes of finite dimensional continuous irreducible representations of K. Then:

(i) E is uniformly dense in C(K), we have the decomposition:

L2(K) = L2 ⊕ [φ]∈ ˆG Eφ = L2 ⊕ [φ]∈ ˆG Hφ⊗ H∗φ,

and the following set gives an orthonormal basis for L2_(K):

{√

dφφij : i, j = 1, . . . , dφ, [φ]∈ ˆG

}

.

(ii) Each φ∈ [ ˆK]occurs in the right and left regular representations of K with multiplicity dφ.

(iii) For each i = 1, . . . , dφthe subspace ofEφspanned by the i-th row (resp. column) of the matrix (φi,j)

(resp.(φ_i,j)) is invariant under the right (resp. left) regular representation, and the later representation is equivalent to φ there.

1.3.10 Remark. The last statement of the Peter-Weyl theorem allows us to realize V as a subspace of L2_(K).

Pick any ei∈ V , then (iii) shows that φ is isomorphic to the subrepresentation of the right regular

represen-tation, spanned by:

πi,1, . . . πi,dφ ∈ L 2_(K), by the equivalence: _∑ cjej ∈ V 7→ ∑ cjφi,j.

In the same manner, we can realize it as a subrepresentation of the left regular representation, by the

equiv-alence: _∑

cjej ∈ V 7→

∑

cjφj,i.

Furthermore (ii) shows that φ occurs with multiplicty dφin this representation.

The decomposition of L2(K)given by the Peter-Weyl theorem makes it possible to compute Cr∗(K). Let

f ∈ C(K), then using proposition 1.1.11:

∥φL(f )∥_B(L1_(G))≤ ∥f∥₁ ≤ ∥f∥∞.

As C(K) is dense in Cr∗(K)andE is dense in C(K) by the Peter-Weyl theorem, it follows that:

C_r∗(K) =

∼

⊕

φ∈ ˆK

Eφ.

Here∼ denotes completion of the direct sum in the reduced group C∗algebra norm defined onB(L2(K)). To compute the K-theory, we use the continuity of the K-theory, see [Wegge-Olsen, 1993, thm. 6.2.9]. We can write write C_r∗(K)as a direct limit over ˆKof C∗-algebras as ˆKis countable and⊕_φ_{∈ ˆ}_KEφis dense in

C_r∗(K). Therefore, by continuity of the K-theory:

K_∗(C_r∗(K)) = ⊕

φ∈ ˆK

K_∗(Eφ) ,

butEφ∼= Mdφ(C) for each φ, so we obtain:

= ⊕

φ∈ ˆK

K_∗(C) .

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1.4. Locally compact abelian groups 19

1.3.11 Proposition. Let K be a compact group. The group C∗algebra is given by:

C∗(K) = C_r∗(K) =

∼

⊕

φ∈ ˆK

Eφ,

and its K-theory by:

K0(Cr∗(K)) =

⊕

φ∈ ˆK

Z, K1(Cr∗(K)) = 0.

1.4. Locally compact abelian groups

Let G be a locally compact abelian group. In this section, an isomorphism Cr∗(G) = C0

( ˆ

G

)

using the Fourier transform is obtained. We follow [Davidson, 1996, ch. VII] for this, the results about the abstact Fourier transform can be found in [Folland, 1995, ch. 4]. First the required results about the abstract Fourier transform are given, then these results are applied to compute C∗(G)and show that C_r∗(G) = C∗(G).

We begin by defining the dual group ˆG, so that we can define the Fourier transformF : L1_(G)_{→ C} 0( ˆG).

For a general group G we denote by ˆGby the set of equivalence classes of irreducible representations. By Schur’s lemma it follows that all irreducible representations of G are 1-dimensional. All the irreducible representations of G can therefore described by its dual group of characters, which is also denoted by ˆG:

1.4.1 Definition. Let G be a locally compact abelian group. Define its dual group ˆGby: ˆ

G :={ξ : G→ S1 ⊂ C : G homomorphism}.

Elements in ξ∈ ˆGare called characters.

We can now define the Fourier transform, which generalises the Fourier transform forRn_{, where c}_Rn₌Rn_,

and ξ(x) = e2πiξx_.

1.4.2 Definition. The Fourier transform is the map:

F : L1_(G)_{→ C} 0( ˆG)

f 7→ ˆfwith ˆf (ξ) =

∫

ξ(x)f (x)dx.

The next proposition shows that the Fourier transform is well defined and defines a∗-homomorphism.

1.4.3 Proposition. The Fourier transform is a norm-decreasing∗-homomorphism from L1(G)to C0

( ˆ

G

) . Its range is a dense subspace of C0

( ˆ

G

) .

1.4.4 Remark. C0( ˆG)is the C∗ algebra of compactly supported functions on ˆGwith pointwise product

and the supremum norm. The Fourier transform thus transforms the convolution product into a pointwise product:

F(f ∗ g) = ˆf· ˆg, f, g ∈ L1(G). The Fourier transform extends to a map L2(G)→ L2

( ˆ

G

)

. The Plancherel theorem shows that this is a unitary isomorphism. This will be used to show that φL: C∗(G)→ Cr∗(G)is an isomorphism.

1.4.5 Theorem (Plancherel theorem). The Fourier transform on L1_(G)_{∩ L}2_(G)_{extends uniquely to a}

uni-tary isomorphism U from L2(G)to L2 (

ˆ

G

) .

We have seen thatF transforms L1_(G)_{into a dense subset of C} 0

( ˆ

G

)

. By computing the norm∥ · ∥C∗(G)

under this isomorphism, it follows that C∗(G) ∼= C0

( ˆ

G

) .

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1.4.6 Proposition. Let G be a locally compact abelian group. The Fourier transformF : L1_(G)_{→ C} 0 ( ˆ G ) extends to an isomorphism: F : C∗_(G)_{−→ C}∼= 0 ( ˆ G ) . Proof. From proposition 1.4.3 it follows thatF(L1(G))is dense in C0

( ˆ

G

)

. The norm in which L1_(G)_is

completed is given by:

∥f∥C∗(G)=sup

{

∥φ(f)∥ : φ is a representation of L1_(G)}_,

But as any representation can be constructed using irreducible representations and ∥φ(f) ⊕ ψ(f)∥ = max (φ(f ), ψ(f )) if ψ, φ are representations, we get:

=sup {

∥ξ(f)∥ : ξ ∈ ˆG

}

.

Now let ξ∈ ˆGand f ∈ L1_{(G), then:}

∥ξ(f)∥ = ∫ ∥f(x)ξ(x)∥dx = ∫ ∥f(x)ξ(x)∥ = ∥ ˆf (ξ)∥, hence: ∥f∥C∗(G)=sup { ∥ ˆf (ξ)∥ : ξ ∈ ˆG } =∥ ˆf∥_∞.

We conclude thatF gives an isometric morphism from(L1_(G),_{∥ · ∥}

C∗(G) ) to ( C0 ( ˆ G ) ,∥ · ∥_∞ ) , so by taking the closure we obtain:

F : C∗_(G)_{−→ C}∼= 0 ( ˆ G ) .

Finally, using the Plancherel theorem, we can show that C_r∗(G) = C∗(G)for abelian groups.

1.4.7 Proposition. Let G be a locally compact abelian group. The left regular representation φL: L1(G)→

B(L2(G))extends to an isomorphism:

φL: C∗(G) ∼

=

−→ Cr∗(G).

Proof. By the theorem 1.4.5, the Fourier transform gives a unitary operator U : L2_(G) _{→ L}2(_Gˆ)_{. If we} describe the left regular representation under this isomorphism we get for f ∈ Cc(G)and g ∈ L2(G)∩

L1(G):

U φλ(f )U∗ˆg = U f∗ g = ˆf ˆg = M_fˆg,ˆ

where M_fˆis the operator which multiplicates pointwise with f . But:

∥M_fˆ∥_B₍_L2₍_Gˆ_{)) = ∥}fˆ∥∞=∥f∥C∗(G),

by [Conway, 2007, thm. 1.5]. We conclude that φLis an isometry on the dense set Cc(G). We have already

shown that the extension of φLto C∗(G)yields a surjective morphism (see equation (1.7)), hence φLextends

to an isomorphism: φL: C∗(G) ∼ = −→ C∗ r(G).

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2. Spin geometry & Dirac operators

In this chapter the Dirac operator is constructed. This is an elliptic differential operator acting on sections of a vector bundleV over a spin Riemannian manifold X. In the first section, the local model for the construction is explained. For each real inner product space V , the Clifford algebra Cl(V ) and its Spin group Spin(V ) are defined. Then Clifford modules are introduced and from the irreducible modules over Cl(V ), the spinor representation is constructed.

Then out of the local model, the Clifford bundle of T X is constructed. To extend the local construction of Clifford modules to bundles of Clifford modules over T X, the existence of a spin structure is required. This is a principal Spin_n-bundle PSpin(X)defined on X. It is shown that every construction from the local model

can by defined globally on X by inducing vector bundles from PSpin(X). It this way the spinor representation

of Spin(V ) gives a spinor bundleS over X.

Finally, the Dirac operator acting on sections ofS is constructed. We show that the Dirac operator can be twisted by a Hermitian vector bundleV over X. In this way a twisted Dirac operator D_V is obtained for each vector bundleV , which gives the general way to construct elliptic operators over a spin manifold X.

2.1. Clifford algebras and the spinor representation

In the section the local model is discussed. We follow [Berline et al., 2004, ch. 3] and [Parthasarathy, 1972, §1]. For the results about the representation theory of the Clifford algebra, see [Lawson and Michelsohn, 1989, §I.5].

Clifford algebras

Let V be a real vector space with inner product Q.

2.1.1 Definition. The real Clifford algebra Cl(V, Q) of (V, Q) is the algebra overR generated by V with the

relations:

v· w + w · v = −2Q(V, W ). (2.1)

The complex Clifford algebra Cl_C(V, Q)of (V, Q) is the algebra overC generated by V with the same relation (2.1). If we have fixed an inner product Q, we simply write Cl(V ) and Cl_C(V ).

2.1.2 Remark. It is possible to construct the Clifford algebra explicitly from the tensor algebra T (V ) =

⊕

n≥0V⊗n. Let I(V, Q) be the two sided ideal generated by:

{v ⊗ v − Q(v, v) · 1 : v ∈ V } .

Then Cl(V, Q) = T (V )/I(V, Q) and Cl_C(V, Q) = T (V ⊗ C) /I (V ⊗ C, Q) ∼=Cl(V, Q)⊗ C.

2.1.3 Remark. In our definition of the Clifford algebra, we used an inner product. The Clifford algebra can

also be defined for a general bilinear symmetric form Q on a real vector space V . For the construction of the Dirac operator in our case, we need the complex Clifford algebra Cl_C(V, Q)of a non-degenerate Q. As each non-degenerate real bilinear form Q generates the same complex Clifford algebra, we assume that Q is an inner product. In chapter 3, we construct the Clifford algebra on T0(G/K), and the inner product naturally

arises from the maximal compact subgroup K.

The Clifford algebra is constructed by taking the quotient of the tensor algebra T (V ) with respect to the ideal I(V, Q). The tensor algebra is graded by the degree of the tensor. In this grading, I(V, Q) consists of elements of even degree. We thus get aZ/2Z grading on the quotient I(V, Q).

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22 2. Spin geometry & Dirac operators

2.1.4 Definition. The Clifford algebra isZ/2Z graded:

Cl(V, Q) = Cl+(V, Q)⊕ Cl−(V, Q),

where Cl+(V, Q)contains the elements of even degree and Cl−(V, Q)contains the elements of odd degree.

Spin groups

The goal of this section is to construct the spin group Spin(V ) and show that it double covers SO(V ). This double cover is required for constructing the spinor representation in the next section.

Let V be a real vector space and fix an inner product Q. Let{e1, . . . , en} be an orthonormal basis of V .

Denote by Cl∗(V )the multiplicative group of invertible elements in Cl(V ). We now construct the spin group Spin(V ) as a subgroup of Cl∗(V ). To define this, we need to have a conjugation on Cl(V ).

2.1.5 Definition. Let x7→ x be the antiautomorphism of Cl(V ) defined on generators by:

v1v2. . . vk7→ (−1)kvk. . . v2v1, v1, . . . vk∈ V.

We can now define the spin group.

2.1.6 Definition. Spin(V ) is the subgroup of Cl∗(V )given by:

Spin(Q) :={x∈ Cl∗(Q) : x∈ Cl+(Q), xex−1∈ V for all e ∈ V, xx = 1}.

We write Spin_nfor the spin group corresponding to V =Rn_{with the standard inner product.}

Spin(Q) can be viewed as as subgroup of Aut(Cl(V )), in this way it becomes a connected Lie group. Using the Carton-Dieudonné theorem, which states that the orthogonal group O(V ) is generated by reflections, one can prove the next result.

2.1.7 Proposition ([Fulton and Harris, 1991, prop. 20.28]). Define ξ : Spin(V )→ Aut(V ) by:

ξ(x)(v) = xvx∗, (2.2)

then ξ is a surjective homomorphism Spin(V )→ SO(V ) with kernel {±1}. It makes Spin(Q) into a con-nected Lie group, double covering SO(V ).

Now let V2be the subspace of Cl(V ) defined as the linear span of:

{ei1ei2 : 1≤ i1 < i2 ≤ n} .

We will show that V2is the Lie algebra of Spin(V ). Define for z ∈ Cl(V ) the map τ(z): Cl(V ) → Cl(V )

by:

τ (z)(x) = [z, x] = zx− xz.

A direct computation shows that for all i, j, k∈ {1, . . . n} with i ̸= j, it holds that:

τ (eiej) (ek) =      0 if k ̸= i, j 2xj if k = i −2xi if k = j.

This shows that τ (z) restricts to an endomorphism V → V if z ∈ V2.

The next result is comparable to the way we can view a matrix Lie group and its Lie algebra as subsets of a matrix algebra. The spin group is constructed as a subset of Cl(V ), and this next lemma shows that also its Lie algebra can be viewed as a subset of Cl(V ). If we have a algebra representation Cl(V ) → End(W ), it induces a Lie group and Lie algebra representation simply by restricting. Finally taking the differential of the double cover ξ : Spin(V )→ SO(V ) gives τ, which yields a Lie algebra isomorphism V2 → so(V ).

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2.1. Clifford algebras and the spinor representation 23

2.1.8 Lemma ([Parthasarathy, 1972, Lemma 1.1]). Identify End(Cl(V )) with the Lie algebra of Aut(V ) and

identify Cl(V ) with a subalgebra of End(Cl(V )) by the left regular representation, so we can see Cl(V ) as a Lie algebra under the commutator bracket. Then:

(i) The Lie algebra of the subgroup Spin(V )⊂ Aut(Cl(V )) is given by V2.

(ii) If W is a vector space and ψ : Cl(V )→ End(W ) is any Clifford algebra representation, the restriction of ψ to Spin(V ) gives a group representation ψ|_{Spin(V )}. The differential of this representation is the Lie algebra representation given by:

d(ψ|_{Spin(V )})= ψ|V2: V2 → End(W ).

(iii) For all g∈ Spin(V ) we have:

ξ(g) = τ (g),

where ξ is the double cover Spin(V )→ SO(V ) given by (2.2). Its differential is given by τ and induces a Lie algebra isomorphism:

(dξ) = τ : V2 → SO(V ) ⊂ End(V ), τ(a)(v) = [a, v].

Spin representations

In this section the spin representations are constructed. First the irreducible representations of Cl_C(V )are described, then the spin representation is obtained by restricting an irreducible representation of Cl(V ) to Spin(V ) to obtain a group representation. Finally, an explicit construction is given for even dimensions of

V.

Let V be a real vector space of dimension n and fix an inner product Q. The next result describes the complex irreducible representations of Cl_C(V ).

2.1.9 Definition. A complex representation of the Clifford algebra Cl(V ) is aC-algebra homomorphism:

c : Cl(V )→ End_C(W ),

where W is a finite dimensional complex vector space. We call W a Cl(V )-module overC, and call the action of Cl(V ) on W Clifford multiplication.

2.1.10 Theorem ([Lawson and Michelsohn, 1989, thm. I.5.7]).

(i) If n is odd there exist up to isomorphism two irreducible complex representation of Cl_C(V ). (ii) If n is even there exists up to isomorphism one irreducible complex representation of Cl_C(V ).

We can now define using lemma 2.1.8 the spin representation:

2.1.11 Definition. The complex spin representation of Spin(V ) is the homomorphism:

σn:Spin(V )→ Aut (Sn) ,

given by the restriction of an irreducible complex representation Cl_C(V ) → End_C(Sn) to Spin(V ) ⊂

Cl(V )⊂ Cl_C(V ).

2.1.12 Proposition ([Lawson and Michelsohn, 1989, prop. 1.5.15]).

(i) If n is odd, the definition of σndoes not depend on which irreducible representation of ClC(V )is used.

(ii) If n = 2m is even, then there exists a decomposition into two irreducible complex representations of Spin(V ):

Sn= Sn+⊕ Sn−.

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24 2. Spin geometry & Dirac operators

Finally, we want to equip the spinor representation with an inner product to make the representations unitary. Let φ : Cl(V )→ End(W ) be a complex representation of Cl_C(V ). Then it induces by lemma 2.1.8 a representation of Spin(V ). As we have the double cover Spin(V )→ SO(V ), this representation descends up to a sign to SO(V ). As SO(V ) is compact, we can by averaging construct an inner product for which

φ|_{Spin(V )}is unitary. We thus obtain:

2.1.13 Proposition. Let φ : Cl(V ) → End(W ) be a complex representation, and consider the Spin

rep-resentation φ|Spin(V ). Then there exists an inner product on W which makes φ|Spin(V ) into an unitary

representation.

Choose any inner product⟨·, ·⟩ on W , and let u ∈ V be a unit vector. As φ(u2)w =−w for all w ∈ W ,

the inner product ^⟨w, w′⟩ := ⟨uw, uw′⟩ satisfies: ^

⟨uw, uw′_⟩′ _{= ^}_{⟨w, w}′_{⟩, w, w}′ _{∈ W.}

As Spin(V ) acts transitively on unit vectors, we obtain:

2.1.14 Corollary. The inner product which φ|Spin(V ) makes into an unitary representation can be chosen

such that Clifford multiplication by unit vectors e∈ V is orthogonal, i.e.: ⟨

φ(e)w, φ(e)w′⟩=⟨w, w′⟩.

2.1.15 Corollary. If dim V is even and φ is the spinor representation we can construct an inner product for

S+and S−separately. In this way the decomposition of S in S±becomes an orthogonal decomposition:

Sn= Sn+⊕ S−n.

2.1.16 Example. If n = 2m is even, we can describe the spinor representation explicitly. Let V be a real vector

of dimension 2m with inner product Q. The inner product Q extends to a bilinear form Q_Con V ⊗ C. Let

Pbe a polarization of V ⊗ C with respect to Q_C, i.e. a n-dimensional isotropic subspace P of V ⊗ C such that V ⊗ C = P ⊕ P . Define an action ψ of Cl_C(V )on S := ΛP by defining for p∈ P, q ∈ P , x ∈ Λp:

ψ(p)x =√2p∧ x,

ψ(q)x =√2ιq(x),

where ιqis the contraction of x with q, defined by:

ιq: ΛlCn→ Λl−1Cn v1∧ · · · ∧ vp 7→ l ∑ i=1 (−1)i+1Q (vi, q) v1∧ · · · ∧ ˆvi∧ · · · ∧ vp.

A direct computation show that this action satisfies p2 = Q(p, p) = 0 = Q(q, q) = q2 and pq + qp =

−2Q(p, q), so this action extends to an action of ClC(V )on Λp. One can check that this action is faithful,

and by counting dimensions we find that ψ gives an isomorphism:

ψ : Cl_C(V )→ End (S) . As End(S) is simple, it follows that this representation is irreducible.

Furthermore, note that ψ(p), ψ(q) change the degree in ΛP , hence if we restrict the action to Cl+_C, the action preserves the degree. We thus get:

ψ

Cl+_C(V ): ClC(V )

+_{→ End}(_S+)_{⊕ End}(_S−)_,

where S+ _{= Λ}+_{P, S}− _{= Λ}−_P _{are the subspaces of respectively even and odd tensors. Restricting ψ to}

Spin(V )⊂ Cl(V ) ⊂ Cl_C(V )now gives the 2m−1_{-dimensional half-spin representations of Spin(V ):}

Dirac induction for semisimple Lie groups

MSc Mathematical Physics

Master Thesis