Endomorphisms of the 2-adic ring $C^{\ast }$-algebra and its Weyl group

Abstract

In this paper, we establish a one-to-one correspondence between a natural monoid and the endomorphisms of the $2$-adic ring $C^{\ast }$-algebra ${\mathcal{Q}}_2$. As a consequence, we classify endomorphisms of ${\mathcal{Q}}_2$ with prescribed images and derive several criteria for the uniqueness of extensions of endomorphisms of a canonical copy of the Cuntz algebra ${\mathcal{O}}_2$ inside ${\mathcal{Q}}_2$ to endomorphisms of ${\mathcal{Q}}_2$. Moreover, we construct an example of an extendable automorphism of ${\mathcal{O}}_2$ that is not a composition of the flip-flop automorphism, the gauge automorphisms, and inner automorphisms, thereby providing negative answers to certain open questions. Using this explicit construction, we also show that the canonical image of $\mathrm{Aut}({\mathcal{Q}}_2,{\mathcal{O}}_2)$ in the outer automorphism group $\mathrm{Out}({\mathcal{Q}}_2)$ is non-abelian. Finally, we characterize the automorphisms of ${\mathcal{Q}}_2$ that globally preserve $C^{\ast }(u)$, and completely describe the Weyl group $\mathcal{W}({\mathcal{Q}}_2,C^{\ast }(u))$. Consequently, several related open questions are answered affirmatively.