Remark

The sets Pos(G) (resp. Pos$_1(G)$ and Pos$_{\leqslant 1}(G)$) are self-adjoint convex subsets of the involutive algebra $\mathscr{C}_b(G)$. They are closed in the space $\mathscr{C}_b(G)$ endowed with the topology of simple convergence. The set Pos(G) is a convex cone with vertex 0 in the real Banach space $\mathscr{C}_b(G)$.