UHF flows and the flip automorphism

A UHF algebra is a C*-algebra A of the type circle times (infinity)(i=1) Mn -i for some sequence (n(i)) with n(i) greater than or equal to 2, where M-n is the algebra of n x n matrices, while a UHF flow alpha is a flow (or a o ne-parameter automorphism group) on the UHF algebra A obtained as circle ti mes (infinity)(i=1) alpha ((i))(t), where alpha ((i))(t) = Ad e(ithi) for s ome h(i) = h(i)(*) is an element of Mn-i. This is the simplest kind of flow s on the UHF algebra we could think of, yet there seem to have been no atte mpts to characterize the cocycle conjugacy class of UHF flows so that we mi ght conclude, e.g., that the nontrivial quasi-free flows on the CAR algebra axe beyond that class. We give here one attempt, which is still short of w hat we have desired, using the flip automorphism of A circle times A. Our c haracterization for a somewhat restricted class of flows (approximately inn er and absorbing a universal UHF flow) says that the flow alpha is cocycle conjugate to a UHF flow if and only if the flip is approximated by the adjo int action of unitaries which are almost invariant under alpha circle times alpha. 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 suitab le sense. We give a solution to this, as a corollary, for the above-mention ed restricted class of flows. We will also discuss several kinds of flows t o clarify the situation.