Morita equivalence of partial group actions and globalization. (Q2790644)

The authors introduce the notion of a \textit{regular} partial action of a group on an algebra. This is a partial action in the sense of \textit{M. Dokuchaev} and \textit{R. Exel}, [Trans. Am. Math. Soc. 357, No. 5, 1931-1952 (2005; Zbl 1072.16025)], satisfying the property that any finite intersection of its domains coincides with their product.NEWLINENEWLINE Two regular partial actions \(\alpha\) and \(\alpha'\) of \(G\) on algebras \(\mathcal A\) and \(\mathcal A'\) are said to be Morita equivalent (\(\alpha\sim\alpha'\)), if there are a Morita context between the algebras, respecting the domains of the partial actions, and a partial action of \(G\) on the corresponding context algebra \(\mathcal C\) whose restrictions on the isomorphic copies of \(\mathcal A\) and \(\mathcal A'\) in \(\mathcal C\) coincide with \(\alpha\) and \(\alpha'\), respectively. The skew group rings [see loc. cit.] by Morita equivalent regular partial actions are proved to be Morita equivalent.NEWLINENEWLINE A partial action of \(G\) is called globalizable if it is isomorphic to the restriction of a global action of \(G\) on an algebra to some ideal of this algebra. The authors prove that each regular partial action has a Morita equivalent globalizable regular partial action. This leads to the notion of a Morita enveloping action of \(\alpha\): it is an enveloping action [see loc. cit.] of some \(\alpha'\sim\alpha\). It is proved that each \(\alpha\) admits a Morita enveloping action which is unique up to Morita equivalence. Moreover, the skew group rings by \(\alpha\) and its enveloping action are Morita equivalent.NEWLINENEWLINE Let \(\alpha\) and \(\alpha'\) be Morita equivalent regular partial actions of \(G\) on \(\mathcal A\) and \(\mathcal A'\), \(\mathcal R=\mathcal A\rtimes_\alpha G\) and \(\mathcal R'=\mathcal A'\rtimes_{\alpha'}G\) the corresponding skew group rings. Under the assumption that \(\mathcal A\) and \(\mathcal A'\) are algebras with orthogonal local units [see \textit{M. Dokuchaev}, \textit{R. Exel} and \textit{J. J. Simón}, J. Algebra 320, No. 8, 3278-3310 (2008; Zbl 1160.16016)] the authors prove that the isomorphism \(\mathrm{FMat}_X(\mathcal R)\cong\mathrm{FMat}_X(\mathcal R')\) established in [Zbl 1160.16016] can be seen as an isomorphism of \(G\)-graded algebras. As a consequence, two Morita equivalent \(s\)-unital partial actions \(\alpha\) and \(\alpha'\) of \(G\) on \(\mathcal A\) and \(\mathcal A'\) extend to isomorphic partial actions of \(G\) on \(\mathrm{FMat}_X(\mathcal R)\) and \(\mathrm{FMat}_X(\mathcal R')\).NEWLINENEWLINE Another interesting result of the paper states that two Morita equivalent \(s\)-unital partial actions of \(G\) on commutative algebras are necessarily isomorphic. The latter also holds for Morita equivalent partial actions of \(G\) on commutative \(C^*\)-algebras.