On Normal Subgroups of Unitary Groups of Some Unital AF-Algebras

In the case of von Neumann factors of types II₁ and $III$, P. de la Harpe proved that, if N is a normal subgroup of the unitary group which contains a non-trivial self-adjoint unitary, then 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 K₀(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 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