Let A be a separable C* algebra. Fix $n\in\mathbb N$.
For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that:
- $A_t \cong \mathcal{O}_n$ (Cuntz algebra).
- Generators of $A_t$ depend continuously in norm on $t$.
Is it true that there are pair of reals $t_1, t_2$ such that $A_{t_1}$ is contained in $A_{t_2}$ ?
Let $A=M_2(\mathcal O_n)$. Fix a nontrivial automorphism $\alpha$ of $\mathcal O_n$, and let $$ A_0=\left\{\begin{bmatrix}a&0\\0&\alpha(a)\end{bmatrix}:\ a\in\mathcal O_n\right\}. $$ Then $A_0\simeq\mathcal O_n$. Now let $$ A_t=u_t\,A_0\,u_t^*, $$ where $$ u_t=\begin{bmatrix}\cos t&\sin t\\ -\sin t&\cos t\end{bmatrix}. $$ The continuity of the generators follows from $$ \|u_txu_t^*-u_sxu_s^*\| \leq2\|x\|\,\|u_t-u_s\|\leq8\|x\|\,|t-s| $$ (using the Mean Value Theorem to estimate $\|u_t-u_s\|$).
Now, suppose that $A_s\subset A_t$. This is $u_sA_0u_s^*\subset u_tA_0u_t^*$, so it implies that $A_r\subset A_0$, where $r=s-t$ (note that $u_su_t=u_{s+t}$). We have $$ A_r=\left\{\begin{bmatrix}a\,\cos^2 r+\alpha(a)\,\sin^2 r&(\alpha(a)-a)\sin r\cos r\\ (\alpha(a)-a)\sin r\cos r&a\,\sin^2r+\alpha(a)\,\cos^2r\end{bmatrix}:\ a\in\mathcal O_n\right\}\subset A_0.$$ In particular, if $0<r<\pi/2$, we obtain from the 1,2 entry that $a=\alpha(a)$ for all $a\in\mathcal O_n$; as $\alpha$ is nontrivial, this is impossible, and it follows that $A_r\not\subset A_0$.