Locally representable one-parameter automophism groups of AF algebras and KMS states
The major problem motivating this paper is the classification of cocycle-conjugacy classes of one-parameter automorphism groups of AF (approximately finite-dimensional) \(C^*\)-algebras. Towards this classification, the author introduces a new invariant, called the top and bottom marginal spectra. Most of the time, the author studies the more specific, but still quite general, situation of a locally representable one-parameter group on an AF algebra. This means that there exists an increasing sequence of finite-dimensional \(C^*\)-subalgebras whose union is dense and such that the one-parameter group leaves invariant all these finite-dimensional \(C^*\)-subalgebras. Another important aspect of a one-parameter group of automorphisms of a \(C^*\)-algebra is the field of KMS states, which is a simplex bundle with base \({\mathbb R}\), as the author, \textit{O. Bratteli} and \textit{G. A. Elliott} introduced in [Yokohama Math. J. 28, 125-167 (1980; Zbl 0466.46057) and Ann. Math., II. Ser. 123, 205-263 (1986; Zbl 0602.46067)]. It is shown in the paper that, even in the special case of a locally representable one-parameter group, an enormous variation of these fields of KMS states is possible. Finally, the author studies pure product type actions on the UHF \(C^*\)-algebra, which is an infinite tensor product of \(2 \times 2\) full matrix algebras, and he shows that two pure product type actions are cocycle equivalent if both their bottom marginal spectra equal \({\mathbb R}^+\).
one-parameter groups on \(C^*\)-algebras
scientific article; zbMATH DE number 1507657
scientific article