The Cuntz algebra $O\_n$ is the (universal) C\*-algebra generated by n-isometries $s\_1,...,s\_n$ such that 
$$\sum\_{i=1}^n s\_is\_i^\ast =\mathbf{1}, \ \hbox{and}\ s\_i^\ast s\_j=\delta\_{ij} \mathbf{1}\ (\hbox{for all}\ i,j).$$


For example if $s\_1, s\_2$ are the generators of $O\_2$ then $s\_1^2, s\_1s\_2, s\_2$ generate a copy of $O\_3$ in $O\_2$ (as they satisfy the relations of $O\_3$) as pointed out by Cuntz in page 184 in [Comm. Math. Phys. 57(1977), 173-185](https://projecteuclid.org/euclid.cmp/1103901288). 


Now i wonder if one can generate a copy of $O\_2$ in $O\_3$ by manipulating the generators of $O\_3$?



Embedding of Cuntz algebras $O_2\subseteq O_3$?
The answer is no. According to Lemma 2.1 in (K. Kawamura, Universal algebra of sectors, Int. J. Alg. Comput. 19(3)(2009), 347–371.) we have:


$Hom(O\_m, O\_n)$ is nonempty if and only if $m=(n-1)k+1$ for some $k \geq 1$.


Applying this to the case for $Hom(O\_2, O\_3)$ gives that we must have $2=2k+1$, which is impossible.


