Communications on Stochastic Analysis

Volume 7 | Number 3 Article 7

9-1-2013 A Bochner-type representation of positive definite mappings on the dual of a compact group Herbert Heyer

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Recommended Citation Heyer, Herbert (2013) "A Bochner-type representation of positive definite mappings on the dual of a compact group," Communications on Stochastic Analysis: Vol. 7 : No. 3 , Article 7. DOI: 10.31390/cosa.7.3.07 Available at: https://digitalcommons.lsu.edu/cosa/vol7/iss3/7 Communications on Stochastic Analysis Serials Publications Vol. 7, No. 3 (2013) 459-479 www.serialspublications.com

A BOCHNER-TYPE REPRESENTATION OF POSITIVE DEFINITE MAPPINGS ON THE DUAL OF A COMPACT GROUP

HERBERT HEYER

Abstract. For an arbitrary compact group G with dual space Σ(G) a Bochner-type bijection is established between positive definite mappings on Σ(G) and central bounded measures on G. This bijection defined by the gen- eralized Fourier transformation is based on the comparative study of three kinds of function algebras: the coefficient algebra, the algebra of convergent Fourier series, and the central Fourier algebra of G.

1. Introduction Bochner’s classical theorem on the characterization of continuous positive def- inite functions on the real line as Fourier transforms of bounded measures has been generalized to various algebraic-topological structures such as abelian groups, Gelfand pairs and commutative hypergroups, to name only a few examples. If G is a locally compact abelian group and G∧ is its dual group, then for any continuous positive definite function φ on G∧ there exists a unique bounded measure τ on G such that φ is represented as the Fourier transform of τ. This representation admits analogues provided for the underlying abstract algebraic-topological structure a dual structure can be defined on which some sort of positive definiteness makes sense. Within the category of locally compact groups G, successful attempts have lead to establishing a Bochner representation beyond the abelian case, in particular under the assumption that G is a compact Lie group. Even for this class of groups, for which the set Σ(G) of irreducible unitary representations serves as the dual object, a new version of positive definiteness has to be invented in order to reach the desired representation. See f.e.[6]. Moreover, there is an axiomatic approach to the Bochner property applying positive definite forms on the coefficient algebra of G in place of positive definite functions on Σ(G) ([4]). In the present paper an application of the central Fourier algebra studied in the unpublished Doctoral dissertation of D.R. Beldin [2] enables us to generalize a Bochner type theorem from connected compact Lie groups to arbitrary compact groups.

Received 2013-8-16; Communicated by D. Applebaum. 2010 Mathematics Subject Classification. Primary 22C05; Secondary 43A30, 43A35. Key words and phrases. Central measures and functions, Fourier transform, positive definite mappings, Bochner representation. 459 460 HERBERT HEYER

The inspiration for writing up previous thoughts on the subject going back to the author’s monographs [4] and [5] is due to D. Applebaum who in his upcoming book [1] points out that the harmonic analysis of compact groups still lacks a precise analogue of Bochner’s theorem. The main result to be presented here states that for every positive definite mapping φ on the dual Σ(G) of a compact group G there exists a unique central measure on G such that its Fourier transform ˆ equals φ. The layout of our exposition is determined by the interplay among three types of function algebras. Chapter 2 contains preliminaries from the representation theory of a compact group G. In particular the coefficient algebra F (G) of G and the Fourier–Stieltjes transform are introduced. In Chapter 3, we recall the properties of central measure and functions on G and study the central Fourier mapping. Moreover, we introduce the algebra K(G) of absolutely convergent Fourier series whose spectrum coincides with G. This result is an important tool in approaching a similar identification for the central Fourier algebra F 0(G) of G which is the subject of Chapter 4. This identification is achieved by extending elements of spec(F 0(G)) for elements in spec(K(G)), where the extension procedure is purely Banach-algebraic, similar to but different from an idea in [3], (34.37). Finally, Chapter 5 is devoted to introduc- ing a suitable notion of positive definite mappings on the dual Σ(G) of G and to proving Bochner’s theorem along the lines of its classical predecessor, by applying the spectral properties of F 0(G) in place of L1(G). The necessary arguments are borrowed from [2]. Although it has been attempted to make the paper more easily readable by reproducing well-known facts and referring consequently to the semi- nal monograph [3], the reader is expected to be familiar with the Gelfand theory for commutative Banach algebras and its application to locally compact abelian groups ([8]).

2. Preliminaries from Representation Theory For a multiplicatively written locally compact group G with unit element e the space of continuous functions on G will be denoted by C(G). On G there exists a Haar measure ωG which is unique within a positive factor. ωG is bounded and hence normalizable to 1 if and only if G is compact. M(G) denotes the 1 Banach space of bounded (Radon) measures on G, and L (G, ωG) the norm-closed subspace of ωG-absolutely continuous measures. In fact, M(G) is a Banach ∗- algebra with respect to convolution and conjugation, admitting the Dirac measure εe as multiplicative unit. We denote by B(H) the ∗-algebra of bounded linear operators on a Hilbert space H, with I as the identity operator. From now on we shall assume G to be a compact group. Let Σ(G) and Σ′(G) denote the sets of equivalence classes of irreducible and finite-dimensional (continuous, unitary) representations of G respectively. Since G is compact, Σ(G) ⊂ Σ′(G). Given σ ∈ Σ′(G) the character of σ is defined by

(σ) χσ(x) := χU (σ) (x) := tr U (x) BOCHNER-TYPE REPRESENTATION 461 for all x ∈ G, where U (σ) is a representative of the class σ. It is a basic result of the harmonic analysis of compact groups G that every σ′ ∈ Σ′(G) admits a decomposition σ′ = M(σ, σ′)σ, σ∈Σ(G) where ′ −1 M(σ, σ ) := χσ(x )χσ′ (x)ωG(dx) G stands for the multiplicity of σ in σ′ ([3], (27.30)). Let σ ∈ Σ(G). We choose U (σ) ∈ σ h(σ), . . . , h(σ) with representing Hilbert space Hσ having an orthonormal basis 1 dσ , where dσ = dim σ := dim Hσ. For i, j = 1, . . . , dσ the functions (σ) (σ) (σ) (σ) x → uij (x) := U (x)hj , hi are the coefficient functions of σ. The spaces

(σ) Fσ(G) := uij : i, j = 1, . . . , dσ , (σ ∈ Σ(G)) and

F (G) := Fσ(G) σ∈Σ(G) are independent of the choice of the representative U (σ) of σ. F (G) is a subalgebra of C(G) called the coefficient algebra of G. A second basic result of the harmonic analysis of compact groups G is the Peter–Weyl theorem which states that F (G) is dense in C(G) with respect to the uniform norm u, and that the system

(σ) dσuij : σ ∈ Σ(G), i, j = 1, . . . , dσ

2 is an orthonormal basis of L (G, ωG). In order to introduce the Fourier mapping on M(G) we need some preparations on operators on Hilbert spaces. Let {Hi : i ∈ I} be a family of Hilbert spaces Hi of dimension di < ∞ (i ∈ I). The set

E(I) := B(Hi) i I ∈ is a ∗-algebra with respect to the familiar operations. In E(I) one introduces various norms: For E := (Ei)i∈I ∈ E(I) and (ai)i∈I with ai ≥ 1 for all i ∈ I one defines

1 p p i∈I aiEiφ if 1 ≤ p < ∞ Ep := p   sup{Eiφ∞ : i ∈ I} if p = ∞ and consider the subspaces

Ep(I) := {E ∈ E(I): Ep < ∞} (1 ≤ p ≤ ∞). 462 HERBERT HEYER

Here 1 n p p j=1 |uj| if 1 ≤ p < ∞ φp(u1, . . . , un) :=   max {|u1|,..., |un|} if p = ∞ and  Xp := φp(x1, . . . , xu), ∗ 1 where x1, .., xn are the eigenvalues of the positive definite operator |X| := (XX ) 2 . For details see [3], (28.24). Now we return to the given compact group G and its dual Σ(G). For each σ ∈ Σ(G) we choose a representative U (σ) with corresponding Hilbert space Hσ of dimension dσ. Then E (Σ(G)) := B(Hσ) σ∈Σ(G) is a ∗-algebra, and we obtain Banach spaces

Ep (Σ(G)) := E ∈ E Σ(G) : Ep < ∞ , where 1 p p d E if 1 ≤ p < ∞ E := σ σ φp p  σ∈Σ(G)   sup {Eσφ∞ } : σ ∈ Σ(G) } if p = ∞ ∗ In fact, E∞(Σ(G)) becomes a Banach -algebra. For a measure ∈ M(G) the  Fourier(–Stieltjes) transform ˆ : Σ(G) → E(Σ(G)) is given by ˆ(σ)h, k := U¯ (σ)(x)h, k(dx) G 1 for all σ ∈ Σ(G), h, k ∈ Hσ. If := fωG with f ∈ L (G, ωG), then ˆ yields the Fourier transform of f. It is shown in [3], (28.36) that the Fourier mapping

F : M(G) → E∞(Σ(G)) given by F() := ˆ for all ∈ M(G) is a norm-decreasing ∗-isomorphism of Banach ∗-algebras.

3. Central Measures and Functions As before we keep the assumption that G is a compact group. A measure ∈ M(G) is called central if it belongs to the center M z(G) of M(G) which means that ∗ ν = ν ∗ whenever ν ∈ M(G). Theorem 3.1. ([3], (28.48)) For each ∈ M(G) the following statements are equivalent: (i) ∈ M z(G). BOCHNER-TYPE REPRESENTATION 463

(σ) (σ) (σ) (ii) ∗ uij = uij ∗ for any system {uij : i, j = 1, . . . , dσ} of coefficients of σ ∈ Σ(G). (iii) ∗ εa = εa ∗ for all a from a dense subset of G. (iv) Given any σ ∈ Σ(G) there exists an a(τ, σ) ∈ C such that

ˆ(σ) = a(, σ)Idσ . z A function f ∈ C(G) is said to be central provided fωG ∈ M (G). There are equivalences analogous to the ones in Theorem 3.1 for the set Cz(G) of central functions in C(G) in place of M z(G). Centrality of measures and functions on G can also be described as follows: For f ∈ C(G) let 0 −1 f (x) := f(yxy )ωG(dy) G whenever x ∈ G. The mapping P : C(G) → C(G) given by P (f) := f 0 for all f ∈ C(G) is continuous. Given ∈ M(G) we define 0 ∈ M(G) by 0(f) := (f 0) for all f ∈ C(G). Then ∈ M z(G) if and only if = 0. From the fact that for f ∈ C(G), f = f 0 if and only if f(x) = f(yxy−1) holds whenever x, y ∈ G it follows that P is surjective from C(G) onto Cz(G). Now we specify the Fourier transform for measures ∈ M z(G) and functions f ∈ Cz(G). Definition 3.2. For ∈ M z(G) the mapping ˆ : Σ(G) → C given by

˙ (σ) := χσd G σ G for all ∈ Σ( ) is said to be thecentral Fourier transform of . z If f ∈ C (G), then ˆ = fωG defines the central Fourier transform fˆ of f. Theorem 3.3. The central Fourier mapping F˙ : M 2(G) → CΣ(G) given by F˙ () := ˆ˙ for all ∈ M z(G) is injective. In order to prove this theorem we consider the ∗-subalgebra F z(G) := F (G) ∪ Cz(G) of the coefficient algebra F (G) of G and prepare the proof by showing the following z Lemma 3.4. F (G) = {χσ : σ ∈ Σ(G)} 464 HERBERT HEYER

Proof. We first establish the inclusion z {χσ : σ ∈ Σ(G)} ⊂ F (G).

In fact, let σ ∈ Σ(G). Then χσ is continuous. Moreover, χσ is central, since for all x, y ∈ G

−1 (σ) −1 χσ(y xy) = tr U (y xy)

(σ) = tr U (x) = χσ(x). For the reverse inclusion we observe that F z(G) = P (F (G)), where P (f) := f 0 for all f ∈ C(G), hence every element of F z(G) is of the form f 0 for some f ∈ F (G). The mapping P being linear it remains to be shown that (σ) (σ) 0 for each coefficient function uij of σ the function uij is a multiple of χσ. But this follows from the subsequent equalities valid for all x ∈ G: (σ) 0 (σ) −1 uij (x) = uij (yxy )ωG(dy) G dσ (σ) (σ) (σ) −1 = uik (y)ukl (x)ulj (y )ωG(dy) G k,l =1 dσ (σ) 1 = ukl (x)δklδij dσ k,l =1 dσ δij (σ) = ukl (x)δkl dσ k,l =1 δij = χσ(x). dσ 

Proof of Theorem 3.3. Clearly, M z(G) is a linear space, and F˙ is linear on M z(G). Therefore it suffices to show that kerF˙ = {0}. Let ∈ M z(G) be given with ˆ˙ = 0 for all σ ∈ Σ(G). Since is central, we z have to justify that (f) = 0 for all f ∈ C (G). By assumption 0 = ˆ˙ (σ) = (χσ) for σ ∈ Σ(G). Now we apply the Peter-Weyl theorem in order to obtain

u F (G) = C(G). Moreover we know from the Lemma that z F (G) = {χσ : σ ∈ Σ(G)} . Since the mapping P : C(G) → Cz(G) introduced above by P (f) := f 0 for all f ∈ C(G) is continuous and surjective, we obtain

u F z(G) = Cz(G). From this follows that (f) = 0 for all f ∈ Cz(G).  BOCHNER-TYPE REPRESENTATION 465

A further algebra to be applied in the sequel is that of absolutely convergent Fourier series. We follow the approach in [3], §34. Given a measure ∈ M(G) one considers the associated family A := {Aσ ∈ Σ(G)} of coefficient operators defined by

(σ) −1 Aσh, k := U (x )h, k (dx) G ∗ for all h, k in the representing Hilbert space Hσ of Aσ. Clearly, Aσ = Dσˆ(σ) Dσ, where Dσ denotes the conjugation operator for σ ∈ Σ(G) in the sense of (27.25) of [3]. Moreover, Ap = ˆp whenever 1 ≤ p ≤ ∞. The formal expression

(σ) dσtr (AσU ) σ∈Σ(G) is called the Fourier (-Stieltjes) series of , in symbols

(σ) ∼ dσtr (AσU ). σ∈Σ(G) 1 Analogously one introduces the notion of Fourier series of a function f ∈ L (G, ωG). The Fourier series (σ) f ∼ dσtr (AσU ) σ∈Σ(G) 1 of f ∈ L (G, ωG) is said to be absolutely convergent provided

dσAσφ1 < ∞. σ∈Σ(G) 1 Let K(G) denote the set of functions in L (G, ωG) which admit on absolutely convergent Fourier series. For f ∈ K(G) we define ˆ fφ1 := f1, where ˆ ˆ f1 = dσf(σ)φ1 σ∈Σ(G)

= dσAσφ1 < ∞. σ∈Σ(G) K(G) is a linear space, F : K(G) → E1(Σ(G)) is a norm-preserving linear isomor- phism, hence K(G) is a Banach space. Since any f ∈ K(G) with

(σ) f ∼ dσtr (AσU ) σ∈Σ(G) is ωG-almost everywhere equal to the continuous function (σ) dσtr (AσU ), σ∈Σ(G) it can be regarded as an element of C(G). 466 HERBERT HEYER

Theorem 3.5. (i) K(G) is a commutative Banach algebra under pointwise op- erations, having 1 as its unit. (ii) spec(K(G)) = D(K(G)), i.e. every multiplicative linear functional on K(G) is an evaluation functional of the form

f → f(a) = εa(f)

for a fixed a ∈ G. Furnished with the Gelfand topology the set {εa : a ∈ G} is homeomorphic to G. A proof of this theorem is contained in [3], §34. We quote the essential argu- ments. (i) (34.18) Basically it has to be shown that for f, g ∈ K(G) with convergent Fourier series (m) f = dmtr (AmU ) m ≥1 and (n) g = dntr (BnU ) n ≥1 respectively the inequalities

(m) (m) fgφ1 ≤ dmdntr (AmU )tr (BnU )φ1 m n ≥1 ≥1 ≤ fφ1 gφ1 hold, where the crucial estimate is

(m) (m) tr (AmU )tr (BnU )φ1 ≤ Amφ1 Bnφ1 valid for all m, n ≥ 1. (ii) (34.20) Since F is a Banach space isomorphism K(G) → E1(Σ(G)), each bounded linear functional L on K(G) is of the form

f → L(f) := dσtr (fˆ(σ)Fσ), σ∈Σ(G) where

F := {Fσ : σ ∈ Σ(G)} ∈ E∞(Σ(G)) and

sup{Fσφ∞ : σ ∈ Σ(G)} = 1. From the multiplicativity of L one deduces that L is an evaluation functional on

F (G). But F (G) is dense in K(G) with respect to the norm φ1 and L is continuous. Consequently L is an evaluation functional on K(G). So far one knows that a → φ(a) := εa is a one-to-one mapping from G into spec(K(G)). The inclusion K(G) ⊂ C(G) and the definition of the Gelfand topology yield the continuity of φ. As spec(K(G)) is compact (as the structure space of K(G)), φ is in fact a homeomorphism. BOCHNER-TYPE REPRESENTATION 467

4. The Central Fourier Algebra Next to the algebras F (G) and K(G) associated with the given compact group G we shall place special emphasis on the central Fourier algebra F 0(G) of G consisting of all functions f ∈ Cz(G) that admit convergent Fourier series in the sense that

˙ ∗ f = fˆ(σ )χσ σ∈Σ(G) with 0 ˙ f := dσ|fˆ(σ)| < ∞. σ∈Σ(G) In analogy to the respective result for K(G) we have the following

Theorem 4.1. F 0(G) is a commutative Banach *-algebra with respect to the usual pointwise operations, complex conjugation as involution, and 0 as norm.

Proof. In order to show that F 0(G) is a commutative normed *-algebra we restrict ourselves to proving the crucial inequality fg0 ≤ f0g||0 valid for all f, g ∈ F 0(G). At first we note that for f, g ∈ F 0(G), σ ∈ Σ(G)

˙ ˆ˙ ∗ ˙ ∗ ∗ ∗ fg(σ) = f(σ1 )gˆ(σ2 )M(σ, σ1 ⊗ σ2 ). σ1,σ2∈Σ(G) In fact,

ˆ˙ ∗ ˙ ∗ fg = f(σ1 )χσ1 gˆ(σ2 )χσ2 σ1∈Σ(G) σ2∈Σ(G) ˆ˙ ∗ ˙ ∗ = f(σ1 )gˆ(σ2 )χσ1⊗σ2 , σ1,σ2∈Σ(G) hence ˙ ˆ˙ ∗ ˙ ∗ ˙ fg(σ) = f(σ1 )gˆ(σ2 )χσ1⊗σ2 (σ), σ1σ2∈Σ(G) where

˙ χσ1⊗σ2 (σ) = χσ(x)χσ1 ⊗σ2 (x)ωG(dx) G

∗ −1 = χσ (x )χσ1⊗σ2 (x)ωG(dx) G ∗ = M(σ , σ1 ⊗ σ2) ∗ = M(σ, (σ1 ⊗ σ2) ) ∗ ∗ = M(σ, σ1 ⊗ σ2 ). 468 HERBERT HEYER

Now we compute the 0-norm of the product fg:

0 ˙ fg = fg dσ σ∈Σ(G) ˆ˙ ∗ ˙ ∗ ∗ ∗ = f(σ1 )gˆ(σ2 )M(σ, σ1 ⊗ σ2 ) dσ σ∈Σ(G) σ1,σ2∈Σ(G) ˆ˙ ∗ ˙ ∗ ∗ ∗ ≤ f(σ1 ) gˆ(σ2 ) M(σ, σ1 ⊗ σ2 )dσ σ∈Σ(G) σ ,σ ∈Σ(G) 1 2 ˙ fˆ σ∗ g˙ σ∗ d ∗ ∗ = ( 1 ) ˆ( 2 ) σ1 ⊗σ2 σ ,σ ∈Σ(G) 1 2 ˙ ∗ ∗ fˆ σ g˙ σ d ∗ d ∗ = ( 1 ) ˆ( 2 ) σ1 σ2 σ ,σ ∈Σ(G) 1 2 = f0g0.

Finally we have to show that F 0(G) is in fact a Banach∗-algebra. For this purpose we need to provide an embedding of F 0(G) into a Banach∗-algebra L1(Σ(G)) of integrable functions on the discrete dual Σ(G) of G. More precisely, on the power set P(Σ(G)) of Σ(G) one defines a mapping d into R¯ + by

dσ if Σ1 is finite d(Σ ) := σ∈Σ1 1 ∞ otherwise Then (Σ(G), P(Σ(G)), d) is a positive measure space. Let L(Σ(G)) denote the set of complex-valued functions on Σ(G) with finite support, and let L1(Σ(G)) be the set of d-integrable functions on Σ(G). Clearly, L(Σ(G)) is a dense subset of L1(Σ(G)) with respect to the norm of L1(Σ(G)). In L1(Σ(G)) we introduce a multiplication by

(φ × ψ)(σ) := φ(σ1)ψ(σ2)M(σ, σ1 ⊗ σ2) σ1,σ2∈Σ(G) and an involution by

φ∗(σ) := φ(σ∗) for all φ, ψ ∈ L1(Σ(G)), σ ∈ Σ(G). Then L1(Σ(G)) becomes a Banach∗-algebra, ˙ and the mapping f → fˆ from F 0(G) into L1(Σ(G)) is an isometric ∗-isomorphism between algebras. As a consequence we see that F 0(G) is a Banach ∗-algebra. 

Theorem 4.2. F 0(G) coincides with the Banach ∗-algebra Kz(G) of central func- tions in K(G).

Proof. 1. F 0(G) ⊂ Kz(G). BOCHNER-TYPE REPRESENTATION 469

0 1 In fact, let f ∈ F (G). Then f ∈ C(G), hence fǫ L (G, ωG). Applying Theorem 3.1 we obtain ˆ˙ fφ1 = dσf(σ)φ1 σ∈Σ(G)

= dσa(f, σ)Idσ φ1 σ∈Σ(G) 2 = dσ|a(f, σ)| σ∈Σ(G) ˆ˙ 2 |f(σ)| 0 = dσ = f < ∞. dσ σ∈Σ(G) Together with the fact that the functions of F 0(G) are central this implies the stated inclusion.

2. Kz(G) ⊂ F 0(G). Let f ∈ Kz(G). Since f is central, Theorem 3.1 yields ˆ f(σ) = a(f, σ)Idσ for some a(f, σ) ∈ C, all σ ∈ Σ(G). Now consider the function

g := a(f, σ)dσχσ σ∈Σ(G) on G. Applying the formula δστ χˆσ(τ) = Idσ dτ valid for all τ ∈ Σ(G), derived from the Peter-Weyl theorem, one sees thatg ˆ = fˆ, ˙ hence by the injectivity of the Fourier transform that g = f. But now fˆ(σ∗) = a(f, σ)dσ for all σ ∈ Σ(G). Consequently ˆ ∞ > fφ1 = dσf(σ)φ1 σ∈Σ(G) = dσ|dσa(f, σ)| σ∈Σ(G) ˙ ∗ = dσ|fˆ(σ )| σ∈Σ(G) ˙ 0 = dσ|fˆ(σ)| = f σ∈Σ(G) It remains to show the continuity of f. We know from the discussion in Chapter 3 that for {Aσ : σ ∈ Σ(G)} ∈ E1(Σ(G)) the function (σ) x → h(x) := dσtr (AσU (x)) σ∈Σ(G) 470 HERBERT HEYER is continuous on G. Since f ∈ Kz(G),

f = a(f, σ)dσχσ σ∈Σ(G) and ˆ ˆ t f(σ) = a(f, σ)Idσ = f(σ) or t Aσ = fˆ(σ) = fˆ(σ) for all σ ∈ Σ(G), hence {fˆ(σ): σ ∈ Σ(G)} ∈ E1(Σ(G)). Moreover, for all x ∈ G

(σ) h(x) = dσtr (AσU (x)) σ∈Σ(G) (σ) = dσtr (a(f, σ)Idσ U (x)) σ∈Σ(G) (σ) = dσa(f, σ)tr (U (x)) σ∈Σ(G) = dσa(f, σ)χσ(x). σ∈Σ(G) This completes the proof. 

Theorem 4.3. G acts on K(G) as a compact group of continuous automorphisms of G.

Proof. For each x ∈ G consider the mapping ix : G → G given by

−1 ix(y) := xyx for all y ∈ G. Moreover, let the mapping i ∈ G → K(G) be defined by

i(x)(f) := f ◦ ix for all f ∈ K(G), x ∈ G. Then i(x)(f) ∈ K(G). In fact, we have the two equalities

1 i(x)(f) = f ◦ ix ∈ L (G, ωG) and i(x)(f)φ1 = dσi(x)(f)(σ)φ1 σ∈Σ(G) ˆ = dσf(σ)φ1 σ∈Σ(G)

= fφ1 < ∞. BOCHNER-TYPE REPRESENTATION 471

In order to see the second equality we compute for all σ ∈ Σ(G), h, k ∈ Hσ

(σ) −1 i(x)(f)(σ)h, k = U¯ (y)h, k f(xyx )ωG(dy) G (σ) −1 = U¯ (x yx)h, k f(y)ωG(dy) G (σ) −1 (σ) (σ) = U¯ (x )U¯ (y)U¯ (x)h, k f(y)ωG(dy) G = V¯x(y)h, k f(y)ωG(dy) G (U) = f (σ)h, k ,

(U) where f denotes the Fourier transform of f with respect to Vx ∈ σ. This implies i(x)(f)(σ) = f (U)(σ), hence (U) ˆ i(x)(f)(σ)φ1 = f (σ)φ1 = f(σ)φ1 . x G i x K G K G Next we observe that for every ∈ the mapping ( ): ( ) → ( ) is an automorphism of K(G). Obviously, i(x) is linear and multiplicative, and since

i(x)(f)φ1 = fφ1 for all f ∈ K(G), i(x) is continuous. Finally, the mapping i: G →Aut(K(G)) is an anti-automorphism of groups. In fact, for x, y, z ∈ G and f ∈ K(G) we obtain

(i(x) ◦ i(y)) (f)(z) = (f ◦ ixy)(z)

= f(ixy)(z) = f(xyzy−1x−1) −1 = (f ◦ ix)(yzy )

= (i(x)(f) ◦ iy)(z) = (i(y) ◦ i(y)(f)(z). 

Theorem 4.4. spec(F 0(G) = D(F 0(G)). Proof. In view of Theorem 4.3 F 0(G) = {f ∈ K(G): f = i(x)(f) for all x ∈ G} is a closed subalgebra of K(G). Let L be a non-vanishing multiplicative linear func- tional of F 0(G). From [7], Theorem 1.3.3 we infer that there exists a multiplicative linear extension L′ of L on K(G). But by Theorem 3.3 spec(K(G)) = D(K(G)), hence there exists an a ∈ G such that

′ L (f) = f(a) = εa(f) for all f ∈ F 0(G). This shows the inclusion spec(F 0(G)) ⊂ D(F 0(G)). The other inclusion is trivial.  472 HERBERT HEYER

5. Bochner’s Theorem For the given compact group G we abbreviate by Σ := Σ(G) and Σ′ := Σ′(G) the sets of equivalence classes of irreducible and finite dimensional (continuous unitary) representations of G respectively.

Definition 5.1. A mapping φ:Σ → C is called positive definite (on Σ) if for every N ≥ 1, every sequence {σ1, . . . , σN } in Σ and every sequence {c1, . . . cN } in C

N ∗ cnc¯m M(σ, σn ⊗ σm)φ(σ) ≥ 0. n,m=1 σ ∈Σ We denote by P (Σ) the set of all positive definite mappings on Σ.

Remark 5.2. Any mapping φ :Σ → C can be extended to a mapping φ′ :Σ′ → C by putting φ′(σ′) := M(σ, σ′)φ(σ) σ ∈Σ for all σ′ ∈ Σ′. Consequently, φ :Σ → C is positive definite if and only if for all N ≥ 1, sequences {σ1, . . . , σN } in Σ and {c1, . . . , cN } in C

N ′ ∗ cnc¯mφ (σn ⊗ σm) ≥ 0. n,m=1 Special Case 5.3 If G is abelian, the above definition of positive definiteness coincides with the usual one. Indeed, in this case Σ = G∗, i.e. representations ∗ ∗ −1 ′ σ ∈ Σ are characters χσ ∈ G , and σ = χσ as well as σ ⊗ σ = χσχσ′ whenever ′ σ, σ ∈ Σ. Given φ ∈ P (Σ) we obtain with the choice N ≥ 1, {σ1, . . . , σN } in Σ and {c1, . . . cN } in C

N

∗ cnc¯m χσ(x)χσn⊗σm (x)ωG(dx) φ(σ) n,m=1 σ G ∈Σ N

∗ = cnc¯m δσ,σn⊗σm φ(σ) n,m=1 σ ∈Σ N ∗ = cnc¯mφ(σn ⊗ σm) n,,m=1 N −1 = cnc¯mφ(χnχm ). n,m=1

Properties 5.4 of φ ∈ P (Σ). 5.4.1 φ(I) ≥ 0, where I denotes the trivial one-dimensional representation in Σ. 5.4.2 φ(σ∗) = φ(σ) for all σ ∈ Σ. BOCHNER-TYPE REPRESENTATION 473

Proofs. 5.4.1 Choosing N = 1 and c1 = 1 we obtain

∗ M(σ, σ1 ⊗ σ1 ) φ(σ) ≥ 0 σ ∈Σ for all σ1 ∈ Σ, in particular for σ1 = I φ(I) = M(σ, I ⊗ I∗) φ(σ) ≥ 0. σ ∈Σ

5.4.2 With the choices N = 2, c1 = 1, c2 = c, σ1 = σ0 and σ2 = I we compute

∗ ∗ M(σ, σ0 ⊗ σ0 ) φ(σ) +c ¯ M(σ, σ0 ⊗ I ) φ(σ) σ σ ∈Σ ∈Σ ∗ 2 ∗ + c M(σ, I ⊗ σ0 ) φ(σ) + |c| M(σ, I ⊗ I ) φ(σ) σ σ ∈Σ ∈Σ ∗ ∗ 2 = M(σ, σ0 ⊗ σ0 ) φ(σ) +cφ ¯ (σ0) + cφ(σ0 ) + |c| φ(I) ≥ 0. σ ∈Σ By 5.4.1 the first and the last term of the sum is real and ≥ 0, and hence the sum ∗ C cφ¯ (σ0) + cφ(σ0 ) is real. Since this property holds for all c ∈ , we choose c = 1 ∗ R ∗ R and c = i to obtain φ(σ0) + φ(σ0 ) ∈ as well as iφ(σ0 ) − iφ(σ0) ∈ . But then ∗  =φ(σ0) φ(σ0 ) which was to be shown.

z In the following discussion we shall make use of two more symbols: M+(G) for the set of non-negative measure in M z(G) and P b(Σ) for the set of mappings φ ∈ P (Σ) that are bounded in the sense that there exists a constant cφ ≥ 0 such that

|φ(σ)| ≤ cφdσ for all σ ∈ Σ.

Theorem 5.5. (Bochner) The restriction F˙1 of the central Fourier mapping F˙ to z z b M+(G) provides a bijection from M+(G) onto P (Σ).

Proof. 1. By Theorem 3.3 F˙1 is injective.

z 2. We show that for every ∈ M (G) the central Fourier transform ˆ˙ = F˙1() of belongs to P b(Σ).

2.1 Let N ≥ 1, {σ1, . . . , σN } ⊂ Σ and {c1, . . . , cN } ⊂ C. By Remark 5.2 we may consider the extension ˆ˙ ′ of ˆ˙ to Σ′ which is of the form

′ ′ ˆ˙ (σ ) = χσ′ d G for all σ′ ∈ Σ′. In fact, let σ′ admit the decomposition

′ σ = M(σ1σ)σ. σ ∈Σ 474 HERBERT HEYER

Then ′ χσ′ = M(σ, σ )χσ, σ ∈Σ hence ˆ˙ ′(σ′) = M(σ, σ′)ˆ˙ (σ) σ ∈Σ ′ = M(σ, σ ) χσ d σ G ∈Σ ′ = M(σ, σ )χσ d G σ ∈Σ = χσ′ d. G Now, N ∗ ˙ cnc¯m M(σ, σn ⊗ σm)ˆ(σ) n,m=1 σ ∈Σ N ˙ ′ ∗ = cnc¯mˆ (σn ⊗ σm) n,m=1 N

= cnc¯m χσn⊗χ¯σm d m,n=1 G N

= cnc¯m χσn χ¯σm d m,n=1 G N 2 = cnχσn d ≥ 0. n=1 This shows that ˆ˙ ∈ P (Σ).

2.2 ˆ˙ is bounded and therefore ˆ˙ ∈ P b(Σ).

In fact, the bound cµˆ˙ := fulfills the requirements, since cµˆ˙ ≥ 0 and

ˆ˙ (σ) = χσd ≤ |χσ| d ≤ dσd = dσ G G G whenever σ ∈ Σ. b z 3. We show that given φ ∈ P (Σ) there exists a measure ∈ M+(G) such that φ = ˆ˙ = F˙1(τ).

b 0 3.1 Let φ ∈ P (Σ) be such that |φ(σ)| ≤ cφdσ for all σ ∈ Σ. For every f ∈ F (G) we introduce ˙ Tφ(f) := φ(σ)fˆ(σ). σ ∈Σ BOCHNER-TYPE REPRESENTATION 475

0 0 Then Tφ is a well-defined linear functional on F (G) which is -continuous and satisfies Tφ ≤ cφ.

3.1.1 From the inequalities ˙ |Tφ(f)| = φ(σ)fˆ(σ) σ ∈Σ ˙ ≤ |φ(σ)||fˆ(σ)| σ ∈Σ ˙ ≤ cφdσ|fˆ(σ)| σ ∈Σ ˙ = cφ |fˆ(σ)|dσ σ ∈Σ 0 = cφf < ∞ 0 valid for all f ∈ F (G), follows the well-definedness of Tφ.

3.1.2 The linearity of Tφ is clear.

0 3.1.3 The -continuity of Tφ has already been shown in 3.1.1. It implies that Tφ ≤ cφ.

3.2 Now we introduce a mapping [ , ] := F 0(G) × F 0(G) → C by [f, g] := Tφ(fg¯) for all f, g ∈ F 0(G). Obviously

3.2.1 f → [f, g] is linear for each g ∈ F 0(G).

3.2.2 For all f, g ∈ F 0(G) we have [f, g] = [ g,f], since

[f, g] = Tφ(fg¯)

= Tφ(gf¯) ˙ = φ(σ∗) gf(σ∗) σ ∈Σ ˙ = φ(σ)gf(σ∗) σ∈Σ = =Tφ(gf¯) [ g,f]. 476 HERBERT HEYER

Here Property 5.4.2 of P (Σ) has been applied.

3.2.3 For all f ∈ F 0(G) we have [f, f] ≥ 0 Indeed,

[f, f] = Tφ(ff¯) ˙ = φ(σ)f(f¯σ) σ ∈Σ ˆ˙ ∗ ˆ˙ ∗ ∗ ∗ = φ(σ) f(σ1 )f(σ2 )M(σ, σ1 ⊗ σ2 ) σ σ ,σ ∈Σ 12∈Σ ˆ˙ ∗ ˆ˙ ∗ ∗ = φ(σ) f(σ1 )f(σ2)M(σ, σ1 ⊗ σ2 ) σ σ ,σ ∈Σ 12∈Σ ˆ˙ ˆ˙ ∗ = f(σ1)f(σ2) M(σ, σ1 ⊗ σ2 )φ(σ) ≥ 0, σ ,σ σ 12∈Σ ∈Σ the latter inequality resulting from the positive definiteness of φ.

3.2.4 From 3.2.1 and 3.2.4 follows the Schwarz inequality

|[f, g]|2 ≤ [f, f][g, g] for all f, g ∈ F 0(G), which implies

2 |Tφ(f)| ≤ φ(I)Tφ(ff¯).

This is seen as follows: 1 ∈ F 0(G), and

[1, 1] = Tφ(11)¯ = Tφ(1) = φ(σ)1(ˆ˙ σ) σ ∈Σ = φ(I)1(ˆ˙ σ) = φ(I).

Consequently,

2 2 2 |Tφ(f)| = |Tφ(f1)¯ | = |[f, 1]|

≤ [f, f][1, 1] = φ(I)Tφ(ff¯).

In order to obtain

3.3 the continuity of Tφ in the sup-norm u one applies the inequality

1 1 n 1 2 1+ ++ n 2 n 3.3.1 |Tφ(f)| ≤ (φ(I)) 2 2 (Tφ(h ) 2 valid for all f ∈ F 0(G), where h := ff,¯ hn := hhn−1 and hn is a real-valued function for all n ≥ 1. The inequality is easily shown by induction. BOCHNER-TYPE REPRESENTATION 477

n 3.3.2 Since F 0(G) is a *-algebra, h2 ∈ F 0(G) for all n ≥ 1, and from the 0- continuity of Tφ follows

n n 1 2 2 0 n 0 ≤ Tφ(h ) ≤ cφ(h ) 2 or 1 1 1 n n 1 2 1+ 2 ++ 2n 2 2 0 2n Tφ(f) ≤ φ(I) cφ (h ) for all f ∈ F 0(G). On the other hand, the form of the spectral radius

1 ρ(h) = lim hn0 n n→∞ 1 n 2n 0 2 = lim h = h˜sup n→∞ of h in terms of the Gelfand transform h˜ of h, considered as an element of 0 C(spec(F (G))) with sup-norm sup yields

1 1 1 1 n 2n 2 1+ ++ n 2n 2 0 |Tφ(f)| ≤ lim (φ(I)) 2 2 c h n→∞ φ 2 = φ(I) h˜sup 2 = φ(I) ff¯sup ¯ = φ(I)2f˜f˜ sup 2 ˜ 2 ≤ φ(I) fsup for all f ∈ F 0(G). To finish the proof of 3.3 it remains to verify that

3.3.2 f˜sup = fu. This equality follows from Theorem 4.4, since 0 f˜sup = sup |f(L)| : L ∈ spec(F (G)) 0 = sup |L(f)| : L ∈ spec(F (G)) = sup {|εx(f): x ∈ G} = sup {|f(x)| : x ∈ G} = fu. z 0 3.4 Since F (G) ⊂ F (G), the Peter-Weyl theorem yields the u-density of 0 z F (G) in C (G). Consequently there exists a linear, u-continuous extension ′ z Tφ of Tφ to C (G). Moreover

3.4.1 Tφ is a positive operator. 0 0 Indeed, for all f ∈ F (G) we have Tφ(ff¯) ≥ 0 by 3.2.3. Since F (G) is u- z z dense in C (G), for an arbitrary f ∈ C (G), there exists a sequence (fn)n≥1 in 0 F (G) which u-converges to f. But then (fn, f¯n)n≥1 u converges to ff¯, and ′ z the u-continuity of Tφ on C (G) implies that ′ ¯ ¯ Tφ(fnfn) → Tφ(ff) 478 HERBERT HEYER

′ ¯ ¯ ′ ¯ as n → ∞. From Tφ(fnfn) = Tφ(fnfn) ≥ 0 follows Tφ(ff) ≥ 0. z z Now let f belong to the cone C+(G) of nonnegative functions in C (G) and let z ′ it be of the form f = gg = gg¯ with g ∈ C+(G). Then Tφ(f) ≥ 0.

3.4.2 Finally, the mapping : C(G) → C defined by

′ (f) := Tφ(f) for all f ∈ C(G) is a positive, hence u-continuous linear functional on C(G) z which by the Riesz theorem is represented by a measure ∈ M+(G). The measure ¯ fulfills the requirements of the theorem: For any σ ∈ Σ

˙ ′ ¯ˆ(σ) = ¯(χσ) = =Tφ(χσ) Tφ(χσ) − ′ ˙ ′ = φ(σ )χσ(σ ) σ′ ∈Σ − ′ = φ(σ ) χσχσ∗ dωG σ′∈Σ G − ′ = φ (σ ) δσ′,σ∗ σ′ ∈Σ = =φ(σ∗) φ(σ), and the proof is complete. 

Remark 5.6. The boundedness condition added to the positive definiteness in the statement of Theorem 5.1 can be dropped once the underlying group G is Abelian. In the case G := SU(2), however, one shows that the condition is necessary. In fact, let L be a non-continuous *-homomorphism on F 0(G) and define a mapping φ on Σ(G) by φ(σ) := L(χσ) for all σ ∈ Σ(G). Then φ is positive definite, but given c > 0, the non-continuity of L supplies a σ ∈ Σ(G) satisfying |L(χσ)| > cdσ. Thus φ does not fulfill the boundedness condition.

References

1. Applebaum, D.:Probability on Compact Lie Groups, book to appear with Springer 2014 2. Beldin, D. R. :On some problems in the harmonic analysis of compact groups, unpublished manuscript 1965 3. Hewitt, E. and Ross, K. A.: Abstract Harmonic Analysis II, Springer, Berlin-Heidelberg-New York, 1970 4. Heyer, H.:Dualit¨at Lokalkompakter Gruppen, Lecture Notes in Math. Vol. 150 (1970), Springer, Berlin-Heidelberg-New York 5. Heyer, H.:Probability Measures on Locally Compact Groups, Springer, Berlin-Heidelberg- New York, 1977 6. Lo, J.T-H. and Ng, S-K: Characterizing Fourier series representations of probability distri- butions on compact Lie groups, Siam J. Appl. math. 48 222-228 (1988) 7. Murphy, G. J.: C∗-algebras and Operator Theory, Academic Press, Boston, 1990 8. Rudin, W.:Fourier Analysis on Groups, Interscience Publishers, New York-London, 1962 BOCHNER-TYPE REPRESENTATION 479

Herbert Heyer: Mathematisches Institut Der Universitat¨ Tubingen,¨ Auf Der Mor- genstelle 10, Block C, 72076 Tubingen,¨ Germany E-mail address: [email protected]