Proposition 2

The set Noy$_+(X)$ is a self-adjoint cone in the space $\mathscr{C}(X\times X)$; it is stable under product and it is closed when $\mathscr{C}(X\times X)$ is endowed with the topology of simple convergence.

real numberIt is elementary that if$t\geqslant 0$, and that $\frac{f}{f}\in \in$ NoyNoy$_{++}(X)$.(X), then $tf\in$ Noy$_+(X)$ for every

If $(E_1, g_1)$ and $(E_2, g_2)$ are Hilbertian realizations of universally positive kernels $f_1$ and $f_2$ on X, then the pair $(E_1\oplus E_2, g_1+g_2)$ (resp. the pair $(E_1\widehat{\otimes}_2E_2, g_1\otimes g_2)$) is a Hilbertian realization of $f_1+f_2$ (resp. of $f_1f_2$); these are therefore universally positive kernels.

Characterization (ii) of Noy$_+(X)$ (Theorem 1) implies that this set is closed in $\mathscr{C}(X\times X)$ endowed with the topology of simple convergence.