$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}$As is well known, one can view $\mathbb{CP}^n$ as a quotient of the unit $(2n + 1)$-sphere in $\mathbb{C}^{n+1}$ under the action of $\U(1)$, since every line in $\mathbb{C}^{n+1}$ intersects the unit sphere in a circle. 

Moreover, we have $S^{2n + 1} = \SU(n+1)/\SU(n)$, where $\SU(n)$ embeds into the bottom right-hand corner (say).

My question is: Is there an embedding $j$ of $\U(1)$ into $\SU(n+1)/\SU(n)$ that gives the $\U(1)$ action as a left (right) multiplication, i.e. such that $A.e^{i \theta} = Aj(e^{i \theta})$, for all $A \in \SU(n+1)/\SU(n)$?

For $n=1$, it's easy: $\SU(2) = S^3$, and we embed $e^{i\theta}$ as  
$\left(
  \begin{array}{cc}
   e^{i \theta}  & 0  \\\\
   0             & e^{-i \theta}
  \end{array}
\right)$.