According to Silverman's ''Advanced topics in the arithmetic of elliptic curves'',
Proposition 1.1 Let $E/ \Bbb{C}$ be an elliptic curve with complex multiplication over ring $R⊂\Bbb{C}$. isom $[・]:R \cong End(R)$ which satisfies $[α]^*=αω,α\in R$ exists and unique.
Silverman proves existence by using isom $E_Λ\cong \Bbb{C}$. But I'm stuck with proving uniqueness. From $[α]^*=αω$, $[β]^*=βω$, how can I prove $α=β$ ?