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The Rohlin property for \(\mathbb{Z}^2\)-actions on UHF algebras. (English) Zbl 0947.46044

Let \(m\mathbb{Z}^N= \{(m_1n_1,\dots, m_Nn_N)|(n_1,\dots, n_N)\in \mathbb{Z}^N\}\) for \((m_1,\dots, m_N)\in \mathbb{Z}^N\) and let \(\{\xi_i\}^N_{i= 1}\) be the canonical basis of \(\mathbb{Z}^N\). For \(m,n\in\mathbb{Z}^N\), \(m\geq n\) means \(m_i\geq n_i\) for each \(i\). Let \(\alpha\) be a \(\mathbb{Z}^N\)-action on a unital \(C^*\)-algebra \(A\). Then \(\alpha\) is said to have the Rohlin property if for any vector \(m\in\mathbb{N}^N\) there exist \(R\in\mathbb{N}\) and \(m^{(1)},\dots, m^{(R)}\in \mathbb{N}^N\) with \(m^{(1)},\dots, m^{(R)}\geq m\) and which satisfy the following condition: for any \(\varepsilon (>0)\) and finite subset \(F\) of \(A\), there exist projections \(e^{(r)}_g\) (\(r= 1,\dots, R\), \(g\in \mathbb{Z}^N/m^{(r)} \mathbb{Z}^N\)) in \(A\) satisfying \(\sum_r \sum_g e^{(r)}_g= 1\), \(\|xe^{(r)}_g- e^{(r)}_gx\|< \varepsilon\), \(\|\alpha_{\xi_i}(g^{(r)})- e^{(r)}_{\xi_i+ g} x\|< \varepsilon\) for any \(x\in F\), \(r= 1,\dots, R\), \(i= 1,\dots, N\) and \(g\in \mathbb{Z}^N/m^{(r)} \mathbb{Z}^N\).
An automorphism \(\alpha\) is said to be uniformly outer if for any \(a\in A\), any nonzero projection \(j\in A\) and any \(\varepsilon> 0\), there exist projections \(p_1,\dots, p_n\) in \(A\) such that \(p= \sum p_i\), \(\|p_i a\alpha(p_i)\|< \varepsilon\).
The Rohlin type theorem: Let \(\alpha\) be a \(\mathbb{Z}^2\)-action on a UHF algebra \(A\), then the following conditions are equivalent: (1) \(\alpha\) has the Rohlin property, (2) \(\alpha_g\) is uniformly outer for \(g\in\mathbb{Z}^2\setminus 0\).
Tree notions of conjugacy for \(\mathbb{Z}^N\)-action on \(C^*\)-algebras are introduced and their relationship is discussed. The product type \(\mathbb{Z}^2\)-action on UHF algebras is considered. The main result of the paper is the classification of product type \(\mathbb{Z}^2\)-action with the Rohlin property on UHF algebras up to outer conjugacy.

MSC:

46L35 Classifications of \(C^*\)-algebras
46L40 Automorphisms of selfadjoint operator algebras
46L80 \(K\)-theory and operator algebras (including cyclic theory)
46L55 Noncommutative dynamical systems
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