A property (T) for C*-algebras

We define a notion of Property (T) for an arbitrary $C^{\ast }$-algebra $A$ admitting a tracial state. We extend this to a notion of Property (T) for the pair $(A,B),$ where $B$ is a $C^{\ast }$-subalgebra of $A\mathrm{.}$ Let $\mathrm{\Gamma }$ be a discrete group and $C_r^{\ast }(\mathrm{\Gamma })$ its reduced algebra. We show that $C_r^{\ast }(\mathrm{\Gamma })$ has Property (T) if and only if the group $\mathrm{\Gamma }$ has Property (T) . More generally, given a subgroup $\mathrm{\Lambda }$ of $\mathrm{\Gamma }$, the pair $(C_r^{\ast }(\mathrm{\Gamma }),C_r^{\ast }(\mathrm{\Lambda }))$ has Property (T) if and only if the pair of groups $(\mathrm{\Gamma },\mathrm{\Lambda })$ has Property (T).