Let G G be a locally compact group, and let C ∗ ( G ) C^*(G) and C r ∗ ( G ) C^*_r(G) be the full group C ∗ C^* -algebra and the reduced group C ∗ C^* -algebra of G G . We investigate the relationship between property ( T ) (T) for G G and property ( T ) (T) as well as its strong version for C ∗ ( G ) C^*(G) and C r ∗ ( G ) C^*_r(G) (as defined in [Math. Proc. Cambridge Philos. Soc. 158 (2014), pp. 229–239]). We show that G G has property ( T ) (T) if (and only if) C ∗ ( G ) C^*(G) has property ( T ) (T) . Furthermore, when G G is an [ I N ] [IN] -group, we also prove that G G has property ( T ) (T) if (and only if) C r ∗ ( G ) C^*_r(G) has strong property ( T ) (T) . In the case when G G is second countable and non-compact, we show that property ( T ) (T) and strong property ( T ) (T) of C r ∗ ( G ) C^*_r(G) stands between the non-amenablity of G G and that of its maximal connected subgroup. Hence, if G G is second countable, connected, and non-amenable, then C r ∗ ( G ) C^*_r(G) will have strong property ( T ) (T) . Some of these groups (as for instance G = S L 2 ( R ) G=SL_2(\mathbb {R}) ) do not have property ( T ) (T) .