On fully *-extendable automorphisms of the unitary group of UHF-algebras

Abstract

If $\varphi$ is an automorphism of the unitary group of UHF-algebras and the induced map $\theta_\varphi$ on the projections is an orthoisomorphism, then there exists a $*$-automorphism $\psi$ such that $\psi =\varphi$ on a subgroup containing $K_\infty$. Indeed, if $\varphi$ is a continuous, then $\varphi$ is implemented by a $*$-automorphism of the UHF-algebras. In this paper, we construct automorphisms of the unitary groups of certain UHF-algebras such that no $*$-automorphism coincides with $\varphi$ on the unitary group.