Proposition 8

Let Q(R) be the subgroup of X(T) generated by the roots of G.

a) The centre C(G) of G is a closed subgroup of T, equal to the intersection of the kernels of the roots. The canonical map $X(T/C(G))\rightarrow X(T)$ is injective with image Q(R).

b) The compact group C(G) is isomorphic to the dual of the discrete group $X(T)/Q(R) ($Spectral Theories, Chap. II, §1, no. 1, Def. 2).

c) C(G) reduces to the identity element if and only if Q(R) is equal to X(T).

By §2, no. 2, Cor. 2 of Th. 2, C(G) is contained in T. Since this is the kernel of the adjoint representation, it is the intersection of the kernels of the roots, in other words the orthogonal complement of the subgroup Q(R) of X(T). Thus, the proposition follows from Spectral Theories, Chap. II, §1, no. 7, Th. 4 and no. 5, Th. 2.