Locally constant nonproduct-type cocycles (Q1270005)

Let \({\mathcal A}\) denote an UHF algebra realized as a \(C^\ast\)-algebra \(C^\ast({\mathcal R})\) of continuous functions on a groupoid \({\mathcal R}\), and let \(d\) denote a continuous real-valued 1-cocycle on \({\mathcal R}\) (a continuous homomorphism \(d:{\mathcal R}\to {\mathbb R}\)) implementing an automorphism group \(\alpha^d\) of \({\mathcal A}\) given by pointwise multiplication by the corresponding character. The cocycle is said to be of product type if \(\alpha^d\) is a product type automorphism (regarding \({\mathcal A}\) as an infinite tensor product of full matrix algebras). In case, \(d\) is a locally constant cocycle on the groupoid. On the other hand, the authors construct an example of a \({\mathbb Z}\)-analytic triangular UHF algebra with a locally constant cocycle that is not of product type. The property of being of product type in fact depends on the structure of the ordered Bratteli diagram.