On Normal Subgroups of Unitary Groups of Some Unital ${AF}$-Algebras
DOI:
https://doi.org/10.5644/SJM.03.2.09Keywords:
$UHF$-algebras, dimension group, self-adjoint unitaryAbstract
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
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References
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