Ordered Bratteli diagrams, dimension groups and topological dynamics.

*(English)*Zbl 0786.46053An aim of this paper is to study entire classes of \(C^*\)-algebras \(C(X)\) under a single homeomorphism \(\Phi\) of the compact \(X\) in the case when the “dynamical system” \((X,\Phi)\) is such that \(X\) is zero- dimensional and the transformation \(\Phi\) has a unique minimal set. An important subfamily of such systems is the family of minimal homeomorphisms of Cantor sets. (These systems have a “universal” property.) The requisite invariant for the studied crossed-products is a well-known (cohomological) “dimension group”. It is shown that the order ideal structure relates to minvariant sets and invariant probability measures to states on the group. The actual classification theorem is

Theorem 8.5. Let \((X_ 1,\Phi_ 1,y_ 1)\), \((X_ 2,\Phi_ 2,y_ 2)\) be essentially minimal systems so that \(y_ i\) (\(\in X_ i\)) is not periodic under \(\Phi_ i\) for \(i=1,2\). Then \[ C(X_ 1)\times_{\Phi_ 1} \mathbb{Z} \cong C(X_ 2)\times_{\Phi_ 2} \mathbb{Z} \iff K^ 0 (X_ 1,\Phi_ 1)\approx K^ 0(X_ 2,\Phi_ 2) \] (as ordered groups with distinguished order units.) The relation between Bratteli diagrams and dimension groups is well known; the key to this work is a close study of the diagrams themselves upon which we impose additional structure. The idea of using ordered diagrams to define transformations originated with A. M. Vershik [Soviet Math. Dokl. 24, 97-100 (1981; Zbl 0484.47005)]. The fundamental difference between this work and that of Vershik is that the latter considers his transformations in a measure theoretic context.

Theorem 8.5. Let \((X_ 1,\Phi_ 1,y_ 1)\), \((X_ 2,\Phi_ 2,y_ 2)\) be essentially minimal systems so that \(y_ i\) (\(\in X_ i\)) is not periodic under \(\Phi_ i\) for \(i=1,2\). Then \[ C(X_ 1)\times_{\Phi_ 1} \mathbb{Z} \cong C(X_ 2)\times_{\Phi_ 2} \mathbb{Z} \iff K^ 0 (X_ 1,\Phi_ 1)\approx K^ 0(X_ 2,\Phi_ 2) \] (as ordered groups with distinguished order units.) The relation between Bratteli diagrams and dimension groups is well known; the key to this work is a close study of the diagrams themselves upon which we impose additional structure. The idea of using ordered diagrams to define transformations originated with A. M. Vershik [Soviet Math. Dokl. 24, 97-100 (1981; Zbl 0484.47005)]. The fundamental difference between this work and that of Vershik is that the latter considers his transformations in a measure theoretic context.

Reviewer: V.Deundyak (Rostov-na-Donu)

##### MSC:

46L80 | \(K\)-theory and operator algebras (including cyclic theory) |

46L55 | Noncommutative dynamical systems |

46M20 | Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) |

22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |