Finite group actions on \(C^*\)-algebras with the Rohlin property. I
For a \(C^*\)-algebra \(A\), set \(A^\infty=l^\infty({\mathbb N},A)/c_0(A)\) and \(A_\infty=A^\infty\cap A'\), where \(A\) is identified with the \(C^*\)-subalgebra of constant sequences in \(A^\infty\). A (cocycle) action \(\alpha\) of a finite group \(G\) on a unital \(C^*\)-algebra \(A\) has the Rohlin property if there is a partition of unity \(\{e_g\}_{g\in G}\subset A_\infty\) consisting of projections such that \({\alpha_g}_\infty(e_h)=e_{gh}\), \(g,h\in G\). Let \(A^\alpha\subset A\) be the fixed-point \(C^*\)-subalgebra. In the case when \(A\) is simple and the action \(\alpha\) has the Rohlin property, the author describes the image of the \(K\)-theory map \(K_*(A^\alpha)\to K_*(A)\) induced by the inclusion \(A^\alpha\subset A\). The case of the Cuntz algebra \({\mathcal O}_2\) is considered as an example when \(K\)-groups are trivial, and it is shown that the Rohlin property of an action \(\alpha\) is equivalent to the existence of a \(C^*\)-subalgebra of \(({\mathcal O}_2^\alpha)^\infty\cap{\mathcal O}'_2\) which is isomorphic to \({\mathcal O}_2\). In the case when two actions, \(\alpha\) and \(\beta\), of \(G=\mathbb Z_2\) are outer and strongly approximately inner, it is shown that they are conjugate iff \({\mathcal O}_2^\alpha\) and \({\mathcal O}_2^\beta\) are isomorphic. Model actions of symmetries of \({\mathcal O}_2\) are constructed for given \(K\)-theoretical invariants.
scientific article; zbMATH DE number 2133139
scientific article