EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 15, No. 1, 2022, 249-260 ISSN 1307-5543 – ejpam.com Published by New York Business Global On inductive limit of commutative triples Lamine Timite1,∗, Ibrahima Toure1 1 Department of Mathematics and Informatics, University Felix Houphouet Boigny at Abidjan, Cote d’Ivoire Abstract. In this paper, we extend Olshanski’s work on Gelfand pairs to commutative triples. We introduce the notion of spherical triples as a generalization of commutative triples. We prove that inductive limit of an increasing sequence of commutative triples is a spherical triple which shows that the former is also a generalization of spherical pairs. Furthermore, we define spherical functions associated with these spherical triples. Finally, we characterize these spherical functions by a functional equation and we give some of its properties. 2020 Mathematics Subject Classifications: 43A30, 43A65, 43A90, 22D10 Key Words and Phrases: Commutative triple, Spherical function, Spherical pair, Spherical triple 1. Introduction The notion of Gelfand pairs has been introduced by I. Gelfand in 1950 and developed by many authors. An extension of Gelfand pairs is the notion of commutative triples. The notion of commutative triples has been enough studied by many authors such as: R. Camporesi [8], J. Faraut [1], F. Ricci[9], I. Toure [11], etc.... It has permitted to estab- lish a connection between harmonic analysis on non commutative locally compact groups and the theory of commutative Banach algebra. Indeed the spectrum of a commutative Banach subalgebra in the algebra(for the convolution product) of integrable functions is identified with functions defined on G, called δ-spherical functions which play the same role as exponential functions. This identification has allowed to define in the general case, the δ-spherical Fourier transform and to establish the majority of harmonic analysis results on Rn. In 1980’s, Olshanski ([4], [5], [6] has studied infinite dimensional unitary representations for pairs (G∞,K∞) which are inductive limit of Gelfand pairs (Gn,Kn), where G∞ = ∪∞ n=1Gn and K∞ = ∪∞ n=1Kn. Olshanski has proved that inductive limit of increasing sequence of Gelfand pairs is a spherical pair and has given characterizations of spherical functions associated with these pairs. Some authors have also obtained some ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v15i1.4244 Email addresses: timite.lamine87@ufhb.edu.ci (L. Timite), ibrahima.toure@univ-fhb.edu.ci (I. Toure) http://www.ejpam.com 249 © 2022 EJPAM All rights reserved. L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 250 results about inductive limits of Gelfand pairs. We can mention Vershik[5], S. Kerov[2], J. Faraut [1], R. Marouane[3] etc .... For example in 2007, Rabaoui has proved a Bochner type theorem for pairs (G∞,K∞). In this paper, we extend to commutative triples some Olshanski’s results([4], [5], [6]. In second section, we give some definitions and notations which will be useful for well un- derstanding of this paper. In the last section, we first extend to commutative triples the notion of spherical pairs namely spherical triples and we define spherical function for these triples. Then, we prove that inductive limit of a sequence of commutative triples is a spherical triple. Finally, we characterize spherical functions for these triples by a functional equation and we give some properties of these spherical functions. 2. Preliminaries In this section, we give some notations and definitions for the well-understanding of this paper. Let G be a locally compact group and let K be a compact subgroup of G. G is equipped with a left Haar measure dx and K is equipped with its normalized Haar measure α. Let δ be a unitary irreducible representation of K and let us denote by Eδ the realization space of the representation δ. We put End(Eδ), the space of endomorphisms of Eδ and denote by Cc(G,End(Eδ)) the space of compactly supported continuous functions of G with values in End(Eδ). Cc(G,End(Eδ)) is a convolution algebra where the convolution is defined by: for F,H ∈ Cc(G,End(Eδ)) and x ∈ G, F ∗H(x) = ∫ G F (y−1x)H(y)dy We set, Cc(G,K, δ, δ) := { F ∈ Cc(G,End(Eδ)) : F (kxk′) = δ(k′−1)F (x)δ(k−1)∀k, k′ ∈ K, ∀x ∈ G } , the space of continuous δ-radial functions of G with compact support. Cc(G,K, δ, δ) is a subalgebra of the convolution algebra Cc(G,End(Eδ)). We say that (G,K, δ) is a com- mutative triple if the convolution algebra Cc(G,K, δ, δ) is commutative. If δ is the one dimensional trivial representation then we obtain the classical notion of Gelfand pairs. Let us put χδ := d(δ)ξδ, where d(δ) is the degree of δ and ξδ the character of δ. Let us denote by Ĝ(resp.K̂) the unitary dual of G(resp. K). For U ∈ Ĝ, we denote by mtp(δ, U) the multiplicity of δ in U|K . We know by ([9], theorem 1.1, page 4) that (G,K, δ) is com- mutative if and only if mtp(δ, U) ≤ 1 for all U ∈ Ĝ. Let Ĝ(δ) be the subset of Ĝ consisting of those U ∈ Ĝ that contains δ upon restriction to K. For U ∈ Ĝ(δ) and H its realization space, we designate by H(δ) the isotypic component of δ that is the subspace of vectors which transform under K according to δ. The projection P from H onto H(δ) is defined by: P = ∫ K χδ(k −1)U(k)dk. A function Φ : G −→ End(Eδ) is said to be unitary if ∀g ∈ G,Φ(g)∗ = Φ(g−1), where Φ(g)∗ designates the adjoint of Φ(g). L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 251 3. Spherical triples In this section, we extend to commutative triples some of Olshanski’s results. That will permit us to introduce the notion of spherical triples. We recall first the definition of an admissible representation for the pair (G,K). In fact, if G is a topological Hausdorff group (not necessary localement compact) and K a closed subgroup of G, a unitary representation of G is an admissible representation of the pair (G,K) if its restriction to K is a discrete direct sum of irreducible representations. Definition 1. Let G be a Hausdorff topological group, K be a closed subgroup of G and (δ, Eδ) a unitary irreducible representation of K. (G,K, δ) is a spherical triple if for any unitary irreducible admissible representation (U,H) of G, mtp(δ, U|K) ≤ 1. If mtp(δ, U|K) = 1, U is called a δ-spherical representation. Remark 1. If G is a locally compact group and K is a compact subgroup of G, (G,K, δ) is a spherical triple if and only if (G,K, δ) is a commutative triple. Remark 2. Let us also mention that the notion of spherical triples is a generalization of spherical pairs which corresponds to the case when δ is the one dimensional trivial representation. In fact, if mtp(1K , U|K) ≤ 1 then dimHK ≤ 1, where HK is the space of K-invariant vectors of H. In what follows, we define δ-spherical functions for spherical triples. Let us mention that if mtp(δ, U|K) = 1 then H(δ) is isomorphic to Eδ. Definition 2. Let (G,K, δ) be a spherical triple. A function Φ : G −→ End(Eδ) is a δ-spherical if there exists a δ-spherical representation (U,H) of G such that Φ(g)u = PU(g−1)u,∀g ∈ G, ∀u ∈ Eδ, where P is the orthogonal projection of H onto Eδ. The following theorem gives some properties of δ-spherical functions. Theorem 1. Let Φ : G −→ End(Eδ) be a δ-spherical function. Then i) Φ(e) = I, where I is the identity operator of Eδ. ii) Φ is δ-radial. Proof. Let Φ : G −→ End(Eδ) be a δ-spherical function. i) Then there exits a δ-spherical representation (U,H) of G such that Φ(g)u = PU(g−1)u, ∀g ∈ G, ∀u ∈ Eδ. ∀v ∈ Eδ, Φ(e)v = PU(e)v = Pv = v. Hence Φ(e) = I. L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 252 ii) ∀k1, k2 ∈ K,∀x ∈ G and ∀u ∈ Eδ, Φ(k1xk2)u = PU(k−1 2 x−1k−1 1 )u = PU(k−1 2 )U(x−1)U(k−1 1 )u = U(k−1 2 )PU(x−1)U(k−1 1 )u = δ(k−1 2 )PU(x−1)δ(k−1 1 )u = δ(k−1 2 )Φ(x)δ(k−1 1 )u Hence Φ is a δ-radial function. Now, let G1 ⊆ G2 ⊆ ... ⊆ Gn ⊆ ... be an increasing sequence of locally compact groups such that for each n, Gn is a closed subgroup of Gn+1. We consider again K1 ⊆ K2 ⊆ ... ⊆ Kn ⊆ ... an increasing sequence of compact groups such that for each n, Kn is a compact subgroup of Gn and Kn = Gn ∩Kn+1. The family of pairs (Gn,Kn)n≥1 that we consider, equipped with the system of canonical continuous embeddings tm,n : Gn −→ Gm, n,m ∈ N∗, n ≤ m, constitutes an inductive countable system of topological groups. Hence, we can define the following inductive limit groups : G∞ = ⋃ n≥1 Gn K∞ = ⋃ n≥1 Kn. The topology defined on G∞ is the inductive limit topology. Let (Eδn) be an increasing sequence of Hilbert spaces and for any n ≥ 1, let us consider a unitary representation (δn, Eδn) of Kn. For each n ≥ 1, we consider an isometric embedding in : Eδn −→ Eδn+1 commuting with the action of Kn. Let us put Eδ∞ = ⋃ n∈N∗ Eδn the Hilbert completion of ⋃ n∈N∗ Eδn . Then there exists a unique representation δ∞ of G∞ such that δ∞(k)u = δn(k)u,∀k ∈ Kn,∀u ∈ Eδn . δ∞ is the inductive limit of sequence of representations (δn)n∈N∗ . (G∞,K∞, δ∞) will be called the inductive limit of (Gn,Kn, δn)n∈N∗ . In what follows, we assume that (Gn,Kn, δn) is a commutative triple for each n ∈ N∗. We recall the notion of approximation of irreducible representations for inductive limits. Let us consider (T,H) a unitary representation of G∞ and (Tn,Hn) a sequence of unitary representations of groups Gn. Let us put Σ = {ξ1, ..., ξs} ⊂ H and Σn = {ξ1n, ..., ξsn} ⊂ Hn, n = 1, 2, ... We shall write (Tn,Σn) −→ (T,Σ) if (Tn(g)ξin, ξjn) converges to (T (g)ξi, ξj) uniformly on compact sets in G∞ (1 ≤ i, j ≤ s, g ∈ G∞) .(The reader can refer to [6] for more details) We say that the sequence (Tn) of unitary representations of groups Gn approximates the L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 253 unitary representation T of group G∞ if for any finite subset Σ ⊂ H, it is possible to select finite subsets Σn ⊂ Hn of the same cardinalitiy such that (Tn,Σn) −→ (T,Σ). We know by ([6], theorem 22.9, page 434) that, for any irreducible unitary representation T of the group G∞, there exists a sequence (Tn) of irreducible unitary representations of groups Gn approximating T. In the following theorem, we prove that (G∞,K∞, δ∞) is a spherical triple and we characterize δ∞-spherical functions for (G∞,K∞, δ∞). Theorem 2. i) The inductive limit (G∞,K∞, δ∞) of an increasing sequence of com- mutative triples (Gn,Kn, δn) is a spherical triple. ii) A δ∞-radial unitary function of positive type Φ : G∞ −→ End(Eδ∞) is δ∞-spherical if and only if Φ(e) = I∞ and ∀x, y ∈ G∞,Φ(y)Φ(x) = lim n→∞ ∫ Kn χδn(k)Φ(xky)dαn(k), where I∞ is the identity operator of Eδ∞ . The following lemmas are useful to prove this theorem. Lemma 1. Let (Hn) be an increasing sequence of subspaces of a Hilbert space H. We put: H∞ = ⋃ n≥1 Hn the Hilbert completion of ⋃ n≥1Hn and P : H −→ H∞ the projection of H onto H∞. For any n, we denote by Pn : H −→ Hn the projection of H onto Hn. Then Pn strongly converges to P. Proof. Let v ∈ H, then Pv ∈ H∞. Since ⋃∞ n=1Hn is dense in H∞, there exists a sequence (vm)m ⊂ ⋃∞ n=1Hn such that limm→+∞ ||vm − Pv||H = 0. Since (Hn) is an increasing sequence then for any m ∈ N∗, there exists nm ∈ N∗ such that ∀n ≥ nm, vm ∈ Hn. Let us fix n such that nm ≤ n ≤ m. Since vm ∈ Hn, we have: Pnvm = vm. Hence ||PnPv − Pv||H = ||PnPv − Pnvm + vm − Pv||H ≤ ||PnPv − Pnvm||H + ||vm − Pv||H ≤ (||Pn||+ 1)||Pv − vm||H = 2||Pv − vm||H. If n −→ +∞ then m −→ +∞. Since limm→+∞ ||vm − Pv||H = 0 then limn→+∞ ||PnPv − Pv||H = 0. Hence PnP converges strongly to P. Consequently Pn converges strongly to P because for any n ≥ 1, PnP = Pn. L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 254 Lemma 2. Let Φ : G∞ −→ End(Eδ∞) be a function verifying Φ(e) = I∞ and ∀x, y ∈ G∞,Φ(y)Φ(x) = lim n→∞ ∫ Kn χδn(k)Φ(xky)dαn(k). Then i) I∞ = limn→∞ ∫ Kn χδn(k)Φ(k)dαn(k), where I∞ is the operator identity of Eδ∞ . ii) ∀x ∈ G∞, Φ(x) = lim n→∞ ∫ Kn χδn(k)Φ(kx)dαn(k) = lim n→∞ ∫ Kn χδn(k)Φ(xk)dαn(k) iii) ∀x ∈ G∞, ∀k ∈ K∞, Φ(xk) = Φ(k)Φ(x)and Φ(kx) = Φ(x)Φ(k). Proof. i) I∞ = Φ(e)Φ(e) = limn→∞ ∫ Kn χδn(k)Φ(k)dαn(k) ii) ∀x ∈ G∞, we have: Φ(x) = Φ(e)Φ(x) = lim n→∞ ∫ Kn χδn(k)Φ(xk)dαn(k). In the same way ∀x ∈ G∞,Φ(x) = Φ(x)Φ(e) = lim n→∞ ∫ Kn χδn(k)Φ(kx)dαn(k). iii) ∀k, k′ ∈ Kn, χδn(kk ′) = χδn(k ′k). Consequently ∀x ∈ G∞ and ∀k ∈ K∞, we have: Φ(kx) = Φ(kx)Φ(e) = lim n→∞ ∫ Kn χδn(k1)Φ(k1kx)dαn(k1) = lim n→∞ ∫ Kn χδn(k1k −1)Φ(k1x)dαn(k1) = lim n→∞ ∫ Kn χδn(k −1k1)Φ(k1x)dαn(k1) = lim n→∞ ∫ Kn χδn(k1)Φ(kk1x)dαn(k1) = Φ(x)Φ(k) Φ(xk) = Φ(e)Φ(xk) = lim n→∞ ∫ Kn χδn(k1)Φ(xkk1)dαn(k1) L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 255 = lim n→∞ ∫ Kn χδn(k −1k1)Φ(xk1)dαn(k1) = lim n→∞ ∫ Kn χδn(k1k −1)Φ(xk1)dαn(k1) = lim n→∞ ∫ Kn χδn(k1)Φ(xk1k)dαn(k1) = Φ(k)Φ(x) We come back to the proof of theorem 2. Proof. i) Let (G∞,K∞, δ∞) be the inductive limit of an increasing sequence of commutative triples (Gn,Kn, δn). Let us consider (π,H) a unitary irreducible admissible repre- sentation of G∞ and H(δ∞) the isotypic component of δ∞ in H. Let us assume that mtp(δ∞, π|K∞) > 1. Let ξ1, ξ2 be two non-zero vectors of H(δ∞). Let us denote by Eξ1 = 〈δ∞(k)ξ1, k ∈ K∞〉, the Hilbert completion of vector subspace generated by the set {δ∞(k)ξ1, k ∈ K∞} and Eξ2 = 〈δ∞(k)ξ2, k ∈ K∞〉 the Hilbert completion of vector subspace generated by the set {δ∞(k)ξ2, k ∈ K∞} . Eξ1 and Eξ2 are two distinct copies of Eδ∞ in H. Since (π,H) is a unitary irreducible representation of G∞ then by ([6], theorem 22.9, page 434), there exists a sequence (πn,Hn) of unitary irreducible representations of groups Gn approximating (π,H). Since {ξ1, ξ2} ⊂ H then there exists a sequence {ξn1 , ξn2 } ⊂ Hn such that (πn(k)ξ n i , ξ n j ) −→ n→∞ (π(k)ξi, ξj), ∀k ∈ K∞,∀i, j ∈ {1; 2} uniformly on compact sets. Hence (πn(k)ξ n i , ξ n j ) −→ n→∞ (δ∞(k)ξi, ξj),∀k ∈ K∞, ∀i, j ∈ {1; 2} . In particular, if i = j, we have: (πn(k)ξ n i , ξ n i ) −→ n→∞ (δ∞(k)ξi, ξi). Let us put hni = ξni − PHn(δn)(ξ n i ), where Hn(δn) is the isotypic component of δn in Hn and PHn(δn) is the projection of H onto Hn(δn). Since for any n, (Gn,Kn, δn) is a commutative triple then ∀n,Eδn ' Hn(δn). Since (Eδn) is an increasing sequence of vector spaces then the sequence (Hn(δn)) is also increasing. Then by the lemma 1, PHn(δn) converges strongly to PH∞ , where PH∞ is the projection of H onto H∞ L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 256 and H∞ is the Hilbert completion of ⋃ n≥1Hn(δn). For n sufficiently large, we have (πn(k)h n i , h n i ) = (πn(k)ξ n i , ξ n i )− (πn(k)ξ n i , PHn(δn)(ξ n i ))− (πn(k)PHn(δn)(ξ n i ), ξ n i ) + (πn(k)PHn(δn)(ξ n i ), PHn(δn)(ξ n i )). Then (πn(k)h n i , h n i ) −→ n→∞ (δ∞(k)ξi, ξi)− (π(k)ξi, PH(δ∞)(ξi))− (π(k)PH(δ∞)(ξi), ξi) + (π(k)PH(δ∞(ξi), PH(δ∞)(ξi)). Since PH(δ∞)(ξi) = ξi then (δ∞(k)ξi, ξi)−(π(k)ξi, PH(δ∞)(ξi))−(π(k)PH(δ∞)(ξi), ξi)+ (π(k)PH(δ∞(ξi), PH(δ∞)(ξi)) = 0. Hence (πn(k)h n i , h n i ) −→ n→∞ 0. In particular, if k = e, we have ||hni || −→ n→∞ 0. Therefore for n sufficiently large, ξni is arbitrarily close to PHn(δn)(ξ n i ). So that we can assume for n sufficiently large, ξni ∈ Hn(δn). Let us put En i = 〈πn(k)ξni , k ∈ Kn〉, i = 1, 2, the Hilbert completion of the vector subspace generated by the family {πn(k)ξni , k ∈ Kn} . For any i, En i is copy of Eδn in Hn. Let us suppose that there exists k, k1 ∈ Kn such that ξn = πn(k)ξ n 1 = πn(k1)ξ n 2 . (ξn, ξn) = (πn(k)ξ n 1 , πn(k1)ξ n 2 ) = (πn(k −1 1 k)ξn1 , ξ n 2 ) −→ n→∞ (δ∞(k)ξ1, ξ2) = 0. Consequently ||ξn|| −→ n→∞ 0. So En 1 and En 2 are distinct. It follows that Hn contains two distinct copies of Eδn , which is absurd because (Gn,Kn, δn) is a commutative triple for any n and so the multiplicity of δn in π|Kn is at most 1. ii) Let us assume that Φ is a δ∞-spherical function. Then there exists a δ∞-spherical representation (U,H) of G∞ such that ∀g ∈ G,∀u ∈ Eδ∞ , Φ(g)u = PU(g−1)u, where P is the projection of H onto Eδ∞ . For any n ≥ 1, Pn = ∫ Kn χδn(k −1)U(k)dαn(k) is the projection of H onto Eδn , where αn is the normalized Haar measure on Kn. ∀x, y ∈ G, and ∀v ∈ Eδ, Φ(y)Φ(x)v = PU(y−1)PU(x−1)v = lim n→∞ PU(y−1)PnU(x−1)v = lim n→∞ P ∫ Kn χδn(k −1)U(y−1kx−1)dαn(k)v L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 257 = lim n→∞ ∫ Kn χδn(k −1)PU(y−1kx−1)dαn(k)v = lim n→∞ ∫ Kn χδn(k −1)Φ(xk−1y)dαn(k)v = lim n→∞ ∫ Kn χδn(k)Φ(xky)dαn(k)v Hence ∀x, y ∈ G∞, Φ(y)Φ(x) = lim n→∞ ∫ Kn χδn(k)Φ(xky)dαn(k). Conversely, let us assume that Φ is a δ∞-radial unitary function of positive type verifying Φ(e) = I∞ and limn→∞ ∫ Kn χδn(k)Φ(xky)dαn(k) = Φ(y)Φ(x). Let v ∈ Eδ∞ . Let us put l(x) = (Φ(x)v, v), ∀x ∈ G∞. Since Φ is of positive type, l is also of positive type. Hence there exists a unitary representation (U l,Hl) with a cylic vector ξl such that ∀x ∈ G∞, (Φ(x)v, v) = l(x) = (ξl, U l(x)ξl). We know by the lemma 2 iii), Φ(k1xk2) = Φ(k2)Φ(x)Φ(k1),∀k1, k2 ∈ K∞, ∀x ∈ G∞ and by the lemma 2 i), I∞ = lim n→∞ ∫ Kn χδn(k)Φ(k)dαn(k). So we have: (Φ(x)v, v) = lim n→∞ m→∞ ∫ Kn ∫ Km χδn(k1)χδm(k −1 2 )(Φ(x)Φ(k1)v,Φ(k2)v)dαn(k1)dαm(k2) = lim n→∞ m→∞ ∫ Kn ∫ Km χδn(k1)χδm(k −1 2 )(Φ(k−1 2 )Φ(x)Φ(k1)v, v)dαn(k1)dαm(k2) = lim n→∞ m→∞ ∫ Kn ∫ Km χδn(k1)χδm(k −1 2 )(Φ(k1xk −1 2 )v, v)dαn(k1)dαm(k2) = lim n→∞ m→∞ ∫ Kn ∫ Km χδn(k1)χδm(k −1 2 )(ξl, U l(k1xk −1 2 )ξl)dαn(k1)dαm(k2) = lim n→∞ m→∞ ∫ Kn ∫ Km χδn(k1)χδm(k −1 2 )(U l(k−1 1 ), U l(x)U l(k−1 2 )ξl)dαn(k1)dαm(k2) = lim n→∞ m→∞ (U l(x−1)Pnξ l, Pmξl) = (U l(x−1)Pξl, P ξl) = (PU l(x−1)Pξl, ξl) Therefore we can assume ξl ∈ Hl(δ∞) by changing Hl by the subspace generated by Pξl. Let us put ΦU l (x) = PU l(x−1)P, ∀x ∈ G∞. L. Timite, I. Toure / Eur. J. Pure Appl. Math, 15 (1) (2022), 249-260 258 Hence (Φ(x)v, v) = (ΦU l (x)ξl, ξl), ∀x ∈ G∞ (3.1) Thanks to Lemma 2 ii), we have Φ(k) = δ(k−1), ∀k ∈ K∞. Thus any vector u of E∞ can be expressed as a linear combination ∑ iΦ(ki)v, where ki ∈ K∞,∀i. Let T : Eδ∞ −→ Hl(δ∞) be the function defined by: T (∑ i aiΦ(ki)v ) = ∑ i aiΦ U l (ki)ξ l, where ai ∈ C, and ki ∈ K∞, ∀i. Let us show that T is well-defined. For that we assume that ∑ i aiΦ(ki)v = 0. By the equation (3.1), we have χδn(k)(Φ(ykx)v, v) = χδn(k)(Φ U l (ykx)ξl, ξl),∀x, y ∈ G∞, k ∈ Kn. (3.2) By applying integral and limit on both sides of the equation (3.2), we get (Φ(x)Φ(y)v, v) = (ΦU l (x)ΦU l (y)ξl, ξl), ∀x, y ∈ G∞. Hence (Φ(y)v,Φ(x−1)v) = (ΦU l (y)ξl,ΦU l (x−1)ξl),∀x, y ∈ G∞. (3.3) We deduce that 0 = ( ∑ i aiΦ(ki)v, ∑ j bjΦ(c −1 j )v) = ( ∑ i aiΦ U l (ki)ξ l, ∑ j bjΦ U l (c−1 j )ξl), where cj , ki ∈ K∞ and ai, bj ∈ C; ∀i,∀j. Hence T ( ∑ i aiΦ(ki)v) = ∑ i aiΦ U l (ki)ξ l = 0. Thus T is well defined. T is obviously linear. Now, let’s show that T is bijective. For that we assume that T ( ∑ i aiΦ(ki)v) = ∑ i aiΦ U l (ki)ξ l = 0. Then we claim that∑ i aiΦ(ki)v = 0. By the equation (3.3), we have (Φ(y)v,Φ(x−1)v) = (ΦU l (y)ξl,ΦU l (x−1)ξl),∀x, y ∈ G∞. Hence 0 = ( ∑ i aiΦ(ki)v, ∑ j bjΦ(c −1 j )v) = ( ∑ i aiΦ U l (ki)ξ l, ∑ j bjΦ U l (c−1 j )ξl), where cj , ki ∈ K∞,∀i,∀j. Then ∑ i aiΦ(ki)v = 0. Consequently T is bijective. In other hand, we have (T ∑ i aiΦ(ki)v, T ∑ j bjΦ(cj)v) = ( ∑ i aiΦ U l (ki)ξ l, ∑ j bjΦ U l (cj)ξ l) REFERENCES 259 = ∑ i ∑ j aibj(Φ U l (c−1 j ki)ξ l, ξl) = ∑ i ∑ j aibj(Φ(c −1 j ki)v, v) = ( ∑ i aiΦ(ki)v, ∑ j bjΦ(cj)v) Then T is a unitary linear isomorphism. For any g ∈ G∞, we have TΦ(g) ∑ i aiΦ(ki)v = lim n→∞ ∑ i ai ∫ Kn χδn(k)TΦ(kikg)vdαn(k) = lim n→∞ ∑ i ai ∫ Kn χδn(k)Φ U l (kikg)ξ ldαn(k) = ΦU l (g) ∑ i aiΦ U l (ki)ξ l = ΦU l (g)T ∑ i aiΦ(ki)v, Consequently Φ is unitarily equivalent to ΦU l which is a δ∞-spherical function. Therefore Φ is a δ∞-spherical function. References [1] J. Faraut. 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