# Automorphisms of "rational" Kirchberg algebras

Let $M_{\mathbb{Q}}$ be the universal UHF-algebra and let $\mathcal{O}_{\infty}$ be the infinite Cuntz algebra. Let $A$ be a Kirchberg algebra that satisfies the UCT with $K_0(A) \cong \mathbb{Q}^n$ and $K_1(A) = 0$. By the Kirchberg-Phillips classification theorem the projections $p_i \colon \mathbb{Q}^n \to \mathbb{Q}$ induce homomorphisms $A \otimes \mathbb{K} \to M_{\mathbb{Q}} \otimes \mathcal{O}_{\infty} \otimes \mathbb{K}$, which combine to give a homomorphism $$\psi \colon A \otimes \mathbb{K} \to (M_{\mathbb{Q}} \otimes \mathcal{O}_{\infty} \otimes \mathbb{K})^n$$ that is the identity on $K$-Theory. Since $A \otimes \mathbb{K}$ is simple, $\psi$ is injective.

Given an automorphism $\alpha \in Aut(A \otimes \mathbb{K})$, is there an extension of $\alpha$ to an endomorphism $\beta_{\alpha} \in End((M_{\mathbb{Q}} \otimes \mathcal{O}_{\infty} \otimes \mathbb{K})^n)$ with $\psi \circ \alpha = \beta_{\alpha} \circ \psi$? If yes, can $\beta_{\alpha}$ be chosen such that $\beta_{\alpha \circ \alpha'} = \beta_{\alpha} \circ \beta_{\alpha'}$?

Whether you can choose $\beta$ to behave well under composition may turn out to be hard to prove. You are essentially asking whether actions on $K$-theory lift to actions on the algebra. Even on Kirchberg UCT algebras, this is tricky. In this paper http://arxiv.org/pdf/math/0608093.pdf Katsura does some things for some finite groups.