UHF FLOWS AND THE FLIP AUTOMORPHISM

Abstract

A UHF algebra is a C*-algebra A of the type [Formula: see text] for some sequence (ni) with ni≥2, where Mn is the algebra of n×n matrices, while a UHF flow α is a flow (or a one-parameter automorphism group) on the UHF algebra A obtained as [Formula: see text], where [Formula: see text] for some [Formula: see text]. This is the simplest kind of flows on the UHF algebra we could think of; yet there seem to have been no attempts to characterize the cocycle conjugacy class of UHF flows so that we might conclude, e.g., that the non-trivial quasi-free flows on the CAR algebra are beyond that class. We give here one attempt, which is still short of what we have desired, using the flip automorphism of A⊗A. Our characterization for a somewhat restricted class of flows (approximately inner and absorbing a universal UHF flow) says that the flow α is cocycle conjugate to a UHF flow if and only if the flip is approximated by the adjoint action of unitaries which are almost invariant under α⊗α. Another tantalizing problem is whether we can conclude that a flow is cocycle conjugate to a UHF flow if it is close to a UHF flow in a suitable sense. We give a solution to this, as a corollary, for the above-mentioned restricted class of flows. We will also discuss several kinds of flows to clarify the situation.