Endomorphisms of the Cuntz algebra

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 $$k=n-1$$?

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.

Although not exactly what the OP has in mind, there is another interesting characterization of endomorphisms of $$\mathcal O_n$$ in terms of single elements. Namely there is a one-to-one correspondence between endomorphisms of $$\mathcal O_n$$ and unitary elements of $$\mathcal O_n$$ given as follows:

• If $$u$$ is a unitary element, one defines an endomorphisms $$\varphi _u$$ by sending each generator $$S_i$$ to $$uS_i$$.

• Conversely, given an endomorphism $$\varphi$$, one defines the unitary element $$u_\varphi = \sum_{i=1}^n\varphi (S_i)S_i^*.$$

It is in fact very easy to show that these correspondences are each other's inverse.