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.
Using a generalised Bochner type representation for Olshanski spherical pairs, we establish a Lévy-Khinchin formula for the continuous functions of negative type on the space V ∞ = M (∞, C) of infinite dimensional square complex matrices relatively to the action of the product group K ∞ = U (∞)× U (∞). The space V ∞ is the inductive limit of the spaces V n = M (n, C), and the group K ∞ is the inductive limit of the product groups K n = U (n) × U (n), where U (n) is the unitary group.
In this paper, we study the first-order cohomology space of countable direct limit groups related to Olshanski spherical pairs, relatively to unitary representations which do not have almost invariant vectors. In particular, we prove a variant of Delorme’s vanishing result of the first-order cohomology space for spherical representations of Olshanski spherical pairs.
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