On the Fixed Point Algebra of a UHF Algebra under a Periodic Automorphism of Product Type

Abstract

We study the fixed point algebra ${\mathfrak{A}}^{\alpha }$ of a UHF algebra $\mathfrak{A}$ under a periodic automorphism $\alpha$ of product type. We show an example of ${\mathfrak{A}}^{\alpha }$ which is simple and has more than two tracial states and we characterize the case where ${\mathfrak{A}}^{\alpha }$ has only one tracial state. Next we show that ${\mathfrak{A}}^{\alpha }$ is a UHF algebra if and only if $\mathfrak{A}$ is generated by an infinite family of mutually commuting $\alpha$-invariant type $I_p$ subfactors whose fixed point algebras are abelian and by a UHF subalgebra of ${\mathfrak{A}}^{\alpha }$ which commutes with the former (where $p$ denotes the period of $\alpha$).