A Bochner type theorem for inductive limits of Gelfand pairs

Abstract

In this paper, we prove a generalisation of Bochner-Godement theorem. Our result deals with Olshanski spherical pairs (G, K) defined as inductive limits of increasing sequences of Gelfand pairs (G(n), K(n)) n≥1 . By using the integral representation theory of G. Choquet on convex cones, we establish a Bochner type representation of any element ϕ of the set P ♮ (G) of K-biinvariant continuous functions of positive type on G. extensions of the Bochner theorem had been proved. For example, E. Thoma in 1964 and S. Kerov, G. Olshanski and A. Vershik in 2004 studied the case of the infinite symmetric groupG. Olshanski proved that the inductive limit of an increasing sequence of Gelfand pairs is a spherical pair. Hence, the cited examples and many others are part of G. Olshanski's theory for spherical pairs which was elaborated in 1990 (cf. [16]). However, a Bochner type decomposition in this setting has not been established yet. In this paper, by using Choquet's theorem, we prove such generalisation, answering a question asked by J. Faraut in Infinite Dimensional Harmonic Analysis and Probability (cf. [9]).This paper consists of 4 sections devoted to the following topics : in section 2 we begin by recalling some definitions and results concerning continuous functions of positive type, then we prove that, for a classical Gelfand pair (H, M ), the commutant π ϕ (H)′ is commutative and use this to give a direct proof of the fact that the set P ♮ (H) of M -biinvariant continuous functions of positive type on H is a lattice. In section 3, we move to the general setting of an increasing sequence of Gelfand pairs (G(n), K(n)) n≥1 . Our main tool for establishing the generalised Bochner type decomposition is Choquet's theorem. In order to prove the existence of the decomposition, we embed PFor the uniqueness, we prove that the commutant π ϕ (G)′ remains commutative, and that P ♮ (G) is a lattice too. At the end of this paper, we present some remarks and open questions.

Infinite determinantal measures and the ergodic decomposition of infinite Pickrell measures. I. Construction of infinite determinantal measures

“…The main result of this paper is an explicit description of the measure µ (s) and its identification, after a change of variable, with the infinite Bessel point process considered above. This classification has been obtained by Pickrell [32], [33]; Vershik [45] and Olshanski and Vershik [31] proposed a different approach to this classification in the case of unitarily-invariant measures on the space of infinite Hermitian matrices, and Rabaoui [35], [36] adapted the Olshanski-Vershik approach to the initial problem of Pickrell. In this note, the Olshanski-Vershik approach is followed as well.…”

“…Denote byK (s) n (u 1 , u 2 ) the corresponding n-th Christoffel-Darboux kernel : [44]). As (36) shows, the kernelJ s induces on L 2 ((0, +∞), Leb) the operator of orthogonal projection onto the subspace of functions whose Hankel transform is supported in [0, 1] (see [44]). Proof.…”

Infinite determinantal measures

“…First, we recall the classification of ergodic probability U(∞) × U(∞)-invariant measures on Mat(N, C). This classification has been obtained by Pickrell [21], [22]; Vershik [33] and Olshanski and Vershik [20] proposed a different approach to this classification in the case of unitarily-invariant measures on the space of infinite Hermitian matrices, and Rabaoui [24], [25] adapted the Olshanski-Vershik approach to the initial problem of Pickrell. In this note, the Olshanski-Vershik approach is followed as well.…”

Finiteness of Ergodic Unitarily Invariant Measures on Spaces of Infinite Matrices

“…One expects similar results to hold for all the 10 series of homogeneous spaces (see, e,.g., [4,5]). [6] gave a completely different proof for Pickrell's Classification Theorem of U(∞)-invariant ergodic measures on H, and their method has been adapted to ergodic U(∞) × U(∞)-invariant measures on Mat(N, C) by Rabaoui [10], [11]. The proof of Theorems 1, 2 is based on the Olshanski-Vershik approach.…”