$\displaystyle \varphi:A\rightarrow K(H)\otimes_{*}A,a\mapsto p\otimes a$

show that the above map is a *-homomorphism with A a C*-algebra, H a Hilbert space and p a rank-one projection in K(H).

If we let $\displaystyle \psi_t=\text{Ad}u_t\otimes_{*}\text{Id}_{A}:K(H)\o times_{*}A\rightarrow K(H)\otimes_{*}A$

where $\displaystyle (\text{Ad}u)(x)=uxu^*$ and $\displaystyle u_t=e^{itv}$ with v a self-adjoint operator in B(H)

Show that $\displaystyle \varphi_t=\psi_t\varphi:A\rightarrow K(H)\otimes_{*}A$ is a homotopy