Positive definite distributions on K\GK

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 $\text{C}$¹(G). We show that the corresponding property is no longer true for the space of double cosets $\text{K}\text{G}\text{K}$. If G is of real-rank 1, we construct liner functionals $\text{T}{}_{\mathrm{p}}\text{? (C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{))′}$ for each p, 0 < p ? 2, such that $\text{T}{}_{\mathrm{p}}\text{(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$ but T_p does not extend to a continuous functional on $\text{C}{}^{\mathrm{p}}\text{(K}\text{G}\text{K}\text{)}$. In particular, if p ? 1, T_v does not extend to a continuous functional on $\text{C}{}^1\text{(K}\text{G}\text{K}\text{)}$. 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 $\text{T(f 1 f}{}^1\text{) ? 0, ?f ? C}{}_{\mathrm{c}}{}^{\mathrm{\infty }}\text{(K}\text{G}\text{K}\text{)}$. The main tool used is a theorem of Trombi-Varadarajan.