Consider the Cuntz algebra $\mathcal{O}\_n$ with $n \geq 2$ and let $\text{End}(\mathcal{O}\_n)$ be the set of all (unital) $\ast$-endomorphisms of $\mathcal{O}\_n$. I was wondering if there exists an element $x \in \mathcal{O}\_n$ such that the evaluation map $\text{End}(\mathcal{O}\_n) \rightarrow \mathcal{O\_n},$ $\phi \mapsto \phi (x)$ is injective.


If no, what is the smallest $k \in \mathbb{N}$ for which $x\in \mathcal{O}\_n \otimes \mathbb{C}^k$ exists such that the map $\text{End}(\mathcal{O}\_n) \rightarrow \mathcal{O}\_n \otimes \mathbb{C}^k$ given by $\phi \mapsto (\phi \otimes \mathrm{id}) (x)$ is injective? Is it $k=n-1$?



Endomorphisms of the Cuntz algebra
This is true: $\mathcal O\_n$ is singly generated, i.e. there exists $x\in \mathcal O\_n$ such that $C^\ast(x) = \mathcal O\_n$. In particular, if $\phi, \psi \colon \mathcal O\_n \to B$ are $\ast$-homomorphisms such that $\phi(x) = \psi(x)$, then $\phi = \psi$.


There might be a very direct way of showing this, but here is a proof for $n\geq 2$ ($n\neq \infty$): We will use Theorem 1 from the paper [C. Olsen and W. Zame, Some C\*-algebras with a single generator, Trans. Amer. Math. Soc. 215 (1976), 205–217], which states that if $A$ is a unital $C^\ast$-algebra generated by $k(k+1)/2$ elements of which $k(k-1)/2$ are self-adjoint, then $M\_k(A)$ is singly generated.


Let $s\_1,\dots, s\_n \in \mathcal O\_n$ be the canonical generators. Then $M\_n(\mathcal O\_n) \to \mathcal O\_n$ given by $(a\_{i,j})\_{i,j=1}^n \mapsto \sum\_{i,j=1}^n s\_i a\_{i,j} s\_j^\ast$ is a $\ast$-isomorphism. In particular, $M\_{n^2}(\mathcal O\_n)\cong \mathcal O\_n$, so it suffices to show that $M\_{n^2}(\mathcal O\_n)$ is singly generated.


Note that $\mathcal O\_n$ is generated by $2n$ self-adjoint elements, namely $s\_j + s\_j^\ast$ and $i(s\_j - s\_j^\ast)$ for $j=1,\dots, n$. Taking $A= \mathcal O\_n$ and $k=n^2$ in the above theorem (using that $n^2(n^2-1)/2 \geq 2n$ for $n\geq 2$), it follows that $M\_{n^2}(\mathcal O\_n) \cong \mathcal O\_n$ is singly generated.


Note that the theorem above is constructive, so if one wants, one can write down an explicit single geneator of $M\_{n^2}(\mathcal O\_n)$, and use the isomorphism $M\_n(M\_n(\mathcal O\_n)) \cong M\_n(\mathcal O\_n) \cong \mathcal O\_n$ I described above to express this element in $\mathcal O\_n$.


It is also true for $\mathcal O\_\infty$, but I lack an elementary proof. You can use much deeper machinery, namely that $\mathcal O\_\infty$ is $\mathcal Z$-stable (i.e. $\mathcal O\_\infty \otimes \mathcal Z \cong \mathcal O\_\infty$ where $\mathcal Z$ is the Jiang-Su algebra). Then it follows from the main result of [Thiel, Hannes; Winter, Wilhelm The generator problem for Z-stable C∗-algebras. Trans. Amer. Math. Soc. 366 (2014), no. 5, 2327–2343], which states that any separable, unital $\mathcal Z$-stable $C^\ast$-algebra is singly generated.


