The one-cocycle property for shifts

Abstract

The two-sided shift on the infinite tensor product of copies of the $n\times n$ matrix algebra has the so-called Rohlin property, which entails the one-cocycle property, useful in analyzing cocycle-conjugacy classes. In the case n = 2, the restriction of the shift to the gauge-invariant CAR algebra also has the one-cocycle property. We extend the latter result to an arbitrary $n\geq 2$. As a corollary it follows that the flow $\alpha$ on the Cuntz algebra $\mathcal{O}_n=C^*(s_0,s_1,\dotsc,s_{n-1})$ defined by $\alpha_t(s_j)=e^{ip_jt}s_j$ has the Rohlin property (for flows) if and only if $p_0,\dotsc,p_{n-1}$ generate $\mathbb R$ as a closed sub-semigroup. Note that then such flows are all cocycle-conjugate to each other.