Let [Formula: see text] be the infinite semigroup, inductive limit of the increasing sequence of the semigroups [Formula: see text], where [Formula: see text] is the unitary group of matrices and [Formula: see text] is the semigroup of positive hermitian matrices. The main purpose of this work is twofold. First, we give a complete classification of spherical functions defined on [Formula: see text], by following a general approach introduced by Olshanski and Vershik.10 Second, we prove an integral representation for functions of positive-type analog to the Bochner–Godement theorem, and a Lévy–Khinchin formula for functions of negative type defined on [Formula: see text].

中文翻译：

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Olshanski 球面对半群型

令 [Formula: see text] 是无限半群，半群 [Formula: see text] 递增序列的归纳极限，其中 [Formula: see text] 是矩阵的酉群， [Formula: see text] 是正厄米矩阵半群。这项工作的主要目的是双重的。首先，我们按照 Olshanski 和 Vershik 介绍的一般方法，对 [公式：见正文] 上定义的球函数进行了完整分类。10其次，我们证明了类似于 Bochner-Godement 定理的正型函数的积分表示，以及定义在 [公式：见正文] 上的负型函数的 Lévy-Khinchin 公式。