Property T and strong property T for unital *-homomorphisms

We introduce and study property T and strong property T for unital *-homomorphisms between two unital C^*-algebras. We also consider the relations between property T and invariant subspaces for some canonical unital *-representations. As a corollary, we show that when G is a discrete group, G is nite if and only if G is amenable and the inclusion map i :C_r^{*}\left(G\right)\to \beta \left(l^2\left(G\right)\right) has property T: We also give some new equivalent forms of property T for countable discrete groups and strong property T for unital C^*-algebras.