On Normal Subgroups of Unitary Groups of Some Unital ${AF}$-Algebras - Al-Rawashdeh, Ahmed (Sarajevo Journal of Mathematics, 2024)
In the case of von Neumann factors of types $II_{1}$ and $III$, P. de la Harpe proved that, if $\mathcal{N}$ is a normal subgroup of the unitary group which contains a non-trivial self-adjoint unitary, then $\mathcal{N}$ contains all self-adjoint unitaries of the factor. In this paper, we prove that if $A$ is a unital $AF$-algebra, which is either a $UHF$-algebra or its dimension group $K_0(A)$ is a 2-divisible, then any normal subgroup of the unitary group contains all self-adjoint unitaries if it contains some certain non-trivial self-adjoint unitary. Afterwards, we prove that if two unitary group automorphisms agree on a normal subgroup $\mathcal{N}$ of the unitaries, which contains some certain non-trivial self-adjoint unitary, then they differ by some character on the unitary group of $A$. 2000 Mathematics Subject Classification. 46L05, 46L80, 16U60