## Positive Definite Kernels, Continuous Tensor Products, and Central Limit Theorems of Probability Theory |

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### Contents

Positive definite kernels and continuous tensor products | 1 |

Affine invariant conditionally positive definite kernels | 10 |

Factorisable families of protectively invariant positive | 17 |

Copyright | |

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abelian group additive family affine invariant conditionally borel sets choose and fix compact second countable completes the proof complex numbers complex valued function conditionally positive definite continuous function continuous homomorphism continuous normalised conditionally continuous tensor product converges pointwise current group defined denote disjoint borel sets du(w equation factorisable family family of continuous function on G g,heG geGQ group acting group of complex Haar measure Hence Hilbert space homomorphism identity implies invariant conditionally positive invariant positive definite kernels on XxX Lemma Let G limnsup locally compact second normalised conditionally positive normalised positive definite numbers of modulus positive definite function positive definite kernels projective representation projectively invariant positive Remark representation of G satisfying second countable group second order cocycle sequence space H standard borel space subnet Suppose Theorem topological group topological space uniformly infinitesimal family unitary representation values in H vectors x,yeX x.yeX