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Classification of simple \(C^{\ast}\)-algebras and higher dimensional noncommutative tori. (English) Zbl 1049.46052

After G. A. Elliott’s and G. Gong’s classification of AH-algebras [Ann. Math. (2) 144, 497–610 (1996; Zbl 0867.46041)], it has become very important (“extremely important”, as the author says in the introduction) to obtain a classification theorem for \(C^*\)-algebras that are not inductive limits of finite direct sums of homogeneous \(C^*\)-algebras. The main purpose of this paper is to establish such a classification result which covers at least some of the well-known interesting simple \(C^*\)-algebras that are not known to be AH-algebras and which are important in applications.
In [H.-X. Lin, Proc. Lond. Math. Soc. (3) 83, 199–234 (2001; Zbl 1015.46031)], the author defined a unital simple \(C^*\)-algebra \(A\) which has tracial topological rank zero (written \(TR(A)= 0\)). In the paper under review, he moreover defines the classes \({\mathcal B}{\mathcal D}\) and \({\mathcal L}{\mathcal B}{\mathcal D}\) of \(C^*\)-algebras: A \(C^*\)-algebra \(A\) is said to be in \({\mathcal B}{\mathcal D}\) if there is an integer \(k> 0\) such that every irreducible representation of \(A\) is finite-dimensional and its dimension is no more than \(k\). The integer \(k\) is called the bound. A \(C^*\)-algebra \(A\) is said to be in \({\mathcal L}{\mathcal B}{\mathcal D}\) (locally \({\mathcal B}{\mathcal D}\)) if for any \(\varepsilon> 0\) and any finite subset \({\mathcal F}\subset A\), there exists \(B\in{\mathcal B}{\mathcal D}\) such that \(\text{dist}(x,B)< \varepsilon\) for all \(x\in{\mathcal F}\).
The main result is the following Theorem. Let \(A\) and \(B\) be two unital \(C^*\)-algebras in \({\mathcal L}{\mathcal B}{\mathcal D}\) with \(TR(A)= TR(B)= 0\) satisfying the UCT (Universal Coefficient Theorem). Suppose that there is an order isomorphism \[ \alpha: (K_0(A), K_0(A)_+,[1_A], K_1(A))\to (K(B), K_0(B)_+,[1_B], K_1(B)). \] Then there is an isomorphism \(h: A\to B\) such that \(h_*=\alpha\). This classification result is applied to show that certain simple crossed products are isomorphic so they have the same ordered \(K\)-theory. In particular, irrational higher-dimensional noncommutative tori of the form \(C(\mathbb{T}^k)\times_\theta\mathbb{Z}\) are in fact inductive limits of circle algebras.

MSC:

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