Structure of ideals and isomorphisms of \(C^*\)-crossed products by single homeomorphism
The homeomorphism \(C^*\)-algebra for a topological dynamical system \(\Sigma=(X,\sigma)\), where \(X\) is a compact Hausdorff space (in general, non-metrizable) and \(\sigma\) its homeomorphism, is the crossed product \(A(\Sigma)=C(X)\times_\alpha{\mathbb Z}\) with respect to the action of \({\mathbb Z}\) by powers of \(\alpha\in \Aut(C(X))\), given by \((\alpha f)(x)=f(\sigma^{-1}x)\) [see the author, ``The interplay between topological dynamics and theory of \(C^*\)-algebras'' (1992; Zbl 0892.46070) and \textit{M. Boyle} and the author, J. Math. Soc. Japan 50, No. 2, 317-329 (1998; Zbl 0940.37004)]. The paper studies the ideals and isomorphisms of \(A(\Sigma)\). At first Sect. 2 developes an approximation techniques, involving the convolution with summability kernels on the torus \(T\), for natural Fourier series expansion of the elements of the ``universal'' \(C^*\)-crossed product by \({\mathbb Z}\). Sect. 3 describes the ideals in terms of the kernels of irreducible representations of \(A(\Sigma)\) associated to periodic and aperiodic points of \(\sigma\). A class of ``well-behaved'' ideals, including the kernels of infinite dimensional irreducible representations, is investigated. In Sect. 4 the isomorphisms between \(A(\Sigma_1)\) and \(A(\Sigma_2)\) preserving the mentioned class of ideals are called standard. If the set of periodic points of \(\sigma\) is at most countable every isomorphism through \(A(\Sigma)\) is standard. A general notion of algebraic invariants of \(\Sigma\), i.e., the properties shared by systems with isomorphic \(A(\Sigma)\), is introduced and illustrated (e.g., minimality, topological transitivity, freeness).
scientific article; zbMATH DE number 1523366