International Journal of Analysis and Applications ISSN 2291-8639 Volume 2, Number 1 (2013), 19-25 http://www.etamaths.com FACTS ABOUT THE FOURIER-STIELTJES TRANSFORM OF VECTOR MEASURES ON COMPACT GROUPS YAOGAN MENSAH Abstract. This paper gives an interpretation of the Fourier-Stieltjes trans- form of vector measures by means of the tensor product of Hilbert spaces. It also extends the Kronecker product to some operators arising from the Fourier- Stieltjes transformation and associated with the equivalence classes of unitary representations of a compact group. We obtain among other results the effect of this product on convolution of vector measures. 1. Introduction This paper inspects mainly two things: the Fourier-Stieltjes transformation and the Kronecker product. The importance of the Fourier transformation in math- ematical science and ingeneering, for instance in signal processing, is well known and so we need not to lay emphasis on it. On the other hand, among the ways to bring together matrices there is the Kronecker product. It is extensively used in group theory and physics, specially in quantum information theory to determine for instance exact spin hamiltonian [9], [13]. In quantum physics the quantum states of a system is described by an hermitian positive semi-definite matrix with trace one. If X and Y represent the states of two quantum systems then the Kronecker product X ⊗ Y describes the joint system. Some other fundamental applications of the Kronecker product in signal processing, image processing or quantum com- puting can be found in [7], [11], [2], [10] and [14]. This paper aims to deepen the link between the Kronecker product and the Fourier-Stieltjes transform of vector measures. The rest of the paper is organized as follows. In section 2 we recall the definition of vector measures on compact groups. The section 3 is divided into two parts. In the first part we give an interpretation of the Fourier-Stieltjes transform of vector measures as a bounded vector valued mappings on the tensor product of Hilbert s- paces. The next part extends the Kronecker product to some operators arising from the Fourier-Stieltjes transformation and associated with the equivalence classes of unitary representations of a compact group. Here the effect of this product on the convolution of vector measures is obtained. 2. Preliminaries We summarize the definition of a vector measure on a compact group G following [3] and [6]. We denote by B(G) the σ-field of Borel subsets of the compact group 2010 Mathematics Subject Classification. Primary: 43A77, secondary: 43A30. Key words and phrases. vector measure, Fourier-Stieltjes transform, Kronecker product. c©2013 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 19 20 MENSAH G and by A (in the whole paper) a complex Banach algebra with topological dual A∗. Also denote by 〈x∗,x〉 the duality between A∗ and A. A vector measure is a countably additive set function µ : B(G) → A, that is, for any sequence (An) of pairwise disjoint subsets of B(G), one has (1) µ( ∞ ∪ n=1 An) = ∞∑ n=1 µ(An), where the second member of the above equality is convergent in the norm topology of A. If µ is a vector measure on G then for each x∗ ∈A∗ the measure defined by (2) x∗µ(A) := 〈x∗,µ(A)〉, A ∈ B(G) is a complex measure. Let f be a complex measurable Borel function f defined on G. The function f is said to be µ−integrable if the following two conditions are satisfied: (1) ∀x∗ ∈A∗, f is x∗µ−integrable, (2) ∀A ∈ B(G) , ∃y ∈A, ∀x∗ ∈A∗, 〈x∗,y〉 = ∫ A fdx∗µ. Then we set y = ∫ G f dµ. Let µ : B(G) → A be a vector measure. The semivariation [3] of µ is the nonnegative function |µ| defined on B(G) by (3) |µ|(A) = sup{|x∗µ|(A) : x∗ ∈A,‖x∗‖≤ 1}, ∀A ∈ B(G) where |x∗µ| is the variation of x∗µ. We recall that the variation of a scalar (real or complex) measure m is the extended nonnegative mapping |m| defined on B(G) by (4) |m|(A) = sup π ∑ E∈π |m(E)| where the supremum is taken over all partitions π of A into finite number of pair- wise disjoint members of B(G) [3, page 2]. A vector measure µ is said to be of bounded semivariation if |µ|(G) < ∞. The range of a vector measure is bounded if and only if it is of bounded semivariation [3, page 4]. So a vector measure is said to be bounded if it is of bounded semivari- ation. Denote by M1(G,A) the set of bounded A−valued measures on G. The set M1(G,A) is equipped with the norm (5) ‖µ‖ := ∫ G χGd|µ| = |µ|(G) where χG is the characteristic function of G. Let µ, ν be in M1(G,A). The convolution product of µ with ν is given by : (6) µ∗ν(f) = ∫ ∫ G f(gh)dµ(g)dν(h) ,f ∈C(G,A). where C(G,A) is the set of A−valued continuous functions on G. Equipped with this product and the above norm, the space M1(G,A) is a complex Banach algebra. FACTS ABOUT THE FOURIER-STIELTJES TRANSFORM 21 3. Main results 3.1. Fourier-Stieltjes transformation: a tensor product interpretation. The Fourier(-Stieltjes) transform of a complex valued function (or measure) on a compact group is well-known [5]. For a given function f, the Fourier transform of f is a collection of bounded operators on some Hilbert spaces. In [1], Assiamoua defined the Fourier-Stieltjes transform of a Banach algebra valued measure on a compact group by interpreting it as a collection of some continuous sesquilinear mappings. In this section, we use the modern technique of tensor product to keep the interpretation of the Fourier-Stieltjes transform of a vector measure as a collec- tion of operators. Let Ĝ be the unitary dual of the compact group G i.e. the set of equivalence classes of unitary irreducible representations of G. In any equivalence class σ belonging to Ĝ, we choose an element Uσ and denoted its hilbertian representation space by Hσ. Since G is compact, the Hilbert space Hσ is finite dimensional [4], and we denote its dimension by dσ. We fix definitively a canonical basis (ξ σ 1 , . . . ,ξ σ dσ ) of Hσ. For g ∈ G, we set (7) uσij(g) = 〈U σ g ξ σ j ,ξ σ i 〉 where 〈 , 〉 is the inner product in Hσ, and denote by Uσ the contragredient of the representation Uσ, that is the representation of G on Hσ which matrix elements are given by (8) 〈Uσg ξ σ j ,ξ σ i 〉 = uσij(g), where uσij(g) is the complex conjugate of u σ ij(g). Let us denote by Hσ the conjugate Hilbert space to Hσ. We denote by Hσ⊗̂Hσ the tensor product of Hilbert spaces Hσ ⊗ Hσ equipped with the projective tensor product norm. A basis of Hσ⊗̂Hσ is ( ξσi ⊗ ξ σ j ) 1≤i,j≤dσ . Then the Fourier-Stieltjes transform of a bounded A−valued measure µ is the collection of linear operators (µ̂(σ)) σ∈Ĝ from Hσ⊗̂Hσ into A defined by : (9) µ̂(σ)(ξ ⊗η) = ∫ G 〈Uσg ξ,η〉dµ(g), ξ, η ∈ Hσ. Now, let B(Hσ⊗̂Hσ,A) be the space of bounded linear mappings from Hσ⊗̂Hσ into the Banach algebra A. Let B(Ĝ,A) be the bundle over Ĝ whose fiber at σ is B(Hσ⊗̂Hσ,A), that is (10) B(Ĝ,A) = ∏ σ∈Ĝ B(Hσ⊗̂Hσ,A). For ϕ ∈ B(Ĝ,A), set (11) ‖ϕ‖∞ = sup{‖ϕ(σ)‖ : σ ∈ Ĝ} where ‖ϕ(σ)‖ denotes the operator norm of ϕ(σ). Define (12) B∞(Ĝ,A) = {ϕ ∈ B(Ĝ,A) : ‖ϕ‖∞ < ∞}. Theorem 3.1. For µ ∈ M1(G,A), we have µ̂ ∈ B∞(Ĝ,A). 22 MENSAH Proof. It is clear that for each σ, the object µ̂(σ) is linear from Hσ⊗̂Hσ into A. Now ‖µ̂(σ)(ξ ⊗η)‖ = ‖ ∫ G 〈Uσg ξ,η〉dµ(g)‖ ≤ ∫ G |〈Uσg ξ,η〉|d|µ|(g) ≤ ‖ξ‖‖η‖‖µ‖ = ‖ξ ⊗η‖‖µ‖. Thus ‖µ̂(σ)‖≤‖µ‖ for each σ. Therefore ‖µ̂‖∞ ≤‖µ‖ < ∞. � Let µ ∈ M1(G,A) and σ ∈ Ĝ. We associate with the operator û(σ) the matrix denoted by M(û(σ)) which (i,j)−entry belongs to the Banach algebra A and is given by (13) [M(µ̂(σ))]ij = µ̂(σ)(ξσj ⊗ ξ σ i ). For vector measures µ and ν we denote by (µ̂× ν̂)(σ) the operator from Hσ⊗̂Hσ into A associated with the product of matrices M(ν̂(σ))M(µ̂(σ)), that is (14) (µ̂× ν̂)(σ)(ξσj ⊗ ξ σ i ) = dσ∑ k=1 ν̂(σ)(ξσk ⊗ ξ σ i )µ̂(σ)(ξ σ j ⊗ ξ σ k ). The following theorem is the analogue of the convolution theorem. Theorem 3.2. For µ,ν ∈ M1(G,A), we have µ̂∗ν = µ̂× ν̂. Proof. (̂µ∗ν)(σ)(ξσj ⊗ ξ σ i ) = ∫ G 〈Uσt ξ σ j ,ξ σ i 〉d(µ∗ν)(t) = ∫ ∫ G 〈Uσstξ σ j ,ξ σ i 〉dµ(s)dν(t) = ∫ ∫ G 〈Uσt Uσs ξ σ j ,ξ σ i 〉dµ(s)dν(t). Now, we express Uσs ξ σ j in the canonical basis of Hσ: (15) Uσs ξ σ j = dσ∑ k=1 〈Uσs ξ σ j ,ξ σ k〉ξ σ k . So (̂µ∗ν)(σ)(ξσj ⊗ ξ σ i ) = dσ∑ k=1 ∫ ∫ G 〈Uσs ξ σ j ,ξ σ k〉〈Uσt ξ σ k ,ξ σ i 〉dµ(s)dν(t) = dσ∑ k=1 ∫ G 〈Uσt ξ σ k ,ξ σ i 〉dν(t) ∫ G 〈Uσs ξ σ j ,ξ σ k〉dµ(s) = dσ∑ k=1 ν̂(σ)(ξσk ⊗ ξ σ i )µ̂(σ)(ξ σ j ⊗ ξ σ k ) = (µ̂× ν̂)(σ)(ξσj ⊗ ξ σ i ). � Remark: Application to convolution equations The above result can be useful in the resolution of the convolution equation (16) f ∗h = g FACTS ABOUT THE FOURIER-STIELTJES TRANSFORM 23 where f, g and h (the unknown function) are A-valued functions on G. Here f,g and h can be viewed as the vector measures fdx, gdx and hdx respectively where dx denotes the normalized Haar measure of G. For each σ ∈ Ĝ, the equation (16) is transformed into (17) f̂(σ) × ĥ(σ) = ĝ(σ). Therefore if the operator f̂(σ) is invertible, ĥ(σ) can be derived from (17) and h is recovered by the following reconstruction formula (18) h = ∑ σ∈Ĝ dσ dσ∑ i=1 dσ∑ j=1 ĥ(σ)(ξσj ⊗ ξ σ i )u σ ij. 3.2. The mappings Tσ and their Kronecker product. Now we turn our at- tention over some matrix valued mappings acting on the set of bounded vector measures. Let Mdσ (A) be the set of dσ ×dσ matrices with entries in the Banach algebra A. We denote by Tσ the linear mapping Tσ : M 1(G,A) → Mdσ (A) µ 7→ Tσ(µ) = M(µ̂(σ)). We set ∆(Ĝ) = {Tσ : σ ∈ Ĝ}. The following result shows the effect of each Tσ on convolution of vector measures. Theorem 3.3. For Tσ ∈ ∆(Ĝ), µ,ν ∈ M1(G,A), we have: (19) Tσ(µ∗ν) = Tσ(ν)Tσ(µ). Proof. Tσ(µ∗ν) = M(µ̂∗ν(σ)) = M((µ̂× ν̂)(σ)) according to Theorem 3.2. = M(ν̂(σ))M(µ̂(σ)) = Tσ(ν)Tσ(µ). � Now we are going to extend the Kronecker product to the mappings Tσ. We also give some basic properties of this extension. The following definition of Kronecker product of matrices can be found in [8]. Definition 3.4. Let X = (xij) and Y be two matrices. The Kronecker product (also called tensor product) of X by Y is the matrix (20) X ⊗Y =   x11Y x12Y .. . x1nY x21Y · · · ... ... ... · · · ... xm1Y xm2Y · · · xmnY   . The Kronecker product of matrices is associative, noncommutative and verifies, among other properties, the equality: (21) (X1X2) ⊗ (Y1Y2) = (X1 ⊗Y1)(X2 ⊗Y2). 24 MENSAH where Xi, Yj, i = 1, 2; j = 1, 2 are matrices such that the product X1X2 and Y1Y2 exist; see [8] and [12]. Definition 3.5. Let Tσ1, Tσ2 be in ∆(Ĝ). The Kronecker product of Tσ1 by Tσ2 , denoted by Tσ1 � Tσ2 , is the mapping from M 1(G,A) into Mdσ1dσ2 (A) such that (22) ∀µ ∈ M1(G,A), [Tσ1 � Tσ2 ](µ) = M(µ̂(σ1)) ⊗M(µ̂(σ2)). Remark. Since the Kronecker product of matrices is noncommutative, so is the extended Kronecker product �. Many properties of the Kronecker product of matrices are easily extended to the Kronecker product of the mappings Tσ. As an example we prove the associativity in the following theorem. Theorem 3.6. Let σi, i = 1, 2, 3, be in Ĝ and µ,ν be in M 1(G,A). We have the following associative relation: (23) (Tσ1 � Tσ2 ) � Tσ3 = Tσ1 � (Tσ2 � Tσ3 ). And we have the following effect on the convolution of vector measures: (24) [Tσ1 � Tσ2 ](µ∗ν) = [Tσ1 � Tσ2 ](ν)[Tσ1 � Tσ2 ](µ). Proof. 1. [(Tσ1 � Tσ2 ) � Tσ3 ](µ) = [M(µ̂(σ1)) ⊗M(µ̂(σ2))] ⊗M(µ̂(σ3)). The proof can be completed by using the fact that the Kronecker product of matrices is associative. 2. [Tσ1 � Tσ2 ](µ∗ν) = M(µ̂∗ν(σ1)) ⊗M(µ̂∗ν(σ2)) = [M(ν̂(σ1))M(µ̂(σ1))] ⊗ [M(ν̂(σ2))M(µ̂(σ2))] = [M(ν̂(σ1)) ⊗M(ν̂(σ2))][M(µ̂(σ1)) ⊗M(µ̂(σ2))] = [Tσ1 � Tσ2 ](ν)[Tσ1 � Tσ2 ](µ), using meanwhile the property (21) of Kronecker product of matrices. � 4. Conclusion We interpreted the Fourier-Stieltjes transform of A (Banach algebra)-valued measures on a compact group G as a collection of operators from a tensor product of Hilbert spaces into the Banach algebra A. Therefore it was possible to associate a matrix with each of such operators. Then we extended the Kronecker product of matrices to some matrix valued mappings acting on vector measures. We also computed the effect of these mappings and that of their Kronecker product on the convolution of vector measures. Many other properties could be found if this dis- cussion is deepened. These are the aims of some forthcoming papers. What can be the possible applications to Physics (for example) for joining Kronecker product with Fourier-Stieltjes transform of vector measures? FACTS ABOUT THE FOURIER-STIELTJES TRANSFORM 25 References [1] Assiamoua, V.S.K. and Olubummo, A., Fourier-Stieltjes transforms of vector-valued measures on compact groups, Acta Sci. 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