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

Abstract

We study the fixed point algebra &" of a UHF algebra 21 under a periodic automorphism a of product type. We show an example of $lawhichissimpleandhasmorethantwotracialstatesandwecharacterizethecasewhere91\ast hasonlyonetracialstate.Nextweshowthat$ iff is a UHF algebra if and only if 21 is generated by an infinite family of mutually commuting a-invariant type IP subfactors whose fixed point algebras are abelian and by a UHF subalgebra of 21" which commutes with the former (where p denotes the period of a).