## 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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