Sitaram, Alladi
(1978)
*Positive definite distributions on K\G/K*
Journal of Functional Analysis, 27
(2).
pp. 179-184.
ISSN 0022-1236

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Official URL: http://www.sciencedirect.com/science/article/pii/0...

Related URL: http://dx.doi.org/10.1016/0022-1236(78)90025-3

## Abstract

Let G be a semisimple noncompact Lie group with finite center and let K be a maximal compact subgroup. Then W. H. Barker has shown that if T is a positive definite distribution on G, then T extends to Harish-Chandra's Schwartz space C^{1}(G). We show that the corresponding property is no longer true for the space of double cosets K\G/K. If G is of real-rank 1, we construct liner functionals T_{p}ε(C_{c}^{∞}(K\G/K))' for each p, 0 < p ≤ 2, such that T_{p}(f^{∗}f^{∗})≥0, ∀fε C_{c}^{∞}(K\G/K) but T_{p} does not extend to a continuous functional on C^{p}(K\G/K). In particular, if p≤1, T_{v} does not extend to a continuous functional on C^{1}(K\G/K). We use this to answer a question (in the negative) raised by Barker whether for a K-bi-invariant distribution T on G to be positive definite it is enough to verify that T(f^{∗}f^{∗})≥0, ∀fε C_{c}^{∞}(K\G/K). The main tool used is a theorem of Trombi-Varadarajan.

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ID Code: | 53507 |

Deposited On: | 10 Aug 2011 09:48 |

Last Modified: | 10 Aug 2011 09:48 |

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