##
**Enveloping actions and Takai duality for partial actions.**
*(English)*
Zbl 1036.46037

This paper deals with the problem of deciding whether or not a given partial action is the restriction of some global action, called enveloping action, and the uniqueness of this global action. The author shows that this problem depends on the category under consideration, as much as the definition of a partial action.

In the category of topological spaces with continuous maps, an action \(\beta \) on \(Y\) is an enveloping action of the partial action \(\alpha \) on \(X\), if \(X\) is an open subset of \(Y\), \(\alpha =\beta | _X\), and \(Y\) is the \(\beta \) -orbit of \(X\). He shows that in this category the enveloping action always exists and it is unique up to isomorphism.

In the category of \(C^{*}\)-algebras with their homomorphisms, an action \(\beta \) on \(B\) is an enveloping action of the partial action \(\alpha \) on \(A\), if \(A\) is a closed two-sided ideal of \(B\), \(\alpha =\beta | _A\), and \(B\) is the closed linear \(\beta \) -orbit of \(A\). The author shows that in this category the enveloping action is unique to up isomorphism if it exists, but in general it does not exist. For this reason, the author considers a weaker notion of enveloping action, called Morita enveloping action. To define the notion of Morita enveloping action, he introduces on the set of all partial actions a relation, called Morita equivalence, and studies its properties. An action \(\beta \) on \(B\) is a Morita enveloping action of the partial action \(\alpha \) on \(A\), if \(\beta \) is the enveloping action of a partial action that is Morita equivalent to \(\alpha \). He shows that any partial action has a Morita enveloping action, and this is unique up to Morita equivalence. Moreover, he proves a Takai duality theorem for partial actions: if \(\alpha \) is a partial action of an abelian group \(G\) on a \(C^{*}\)-algebra \(A\), \(\delta \) is the dual coaction of \(G\) on the reduced crossed product \(A\times _{\alpha ,r}G\), and \(\widehat{\delta }\) is the dual action of \(G\) on the reduced crossed product \(\left( A\times _{\alpha ,r}G\right) \times _{\delta ,r}\widehat{G}\) , then \(\widehat{\delta }\) is the Morita enveloping action of \(\alpha \).

As an application, it is shown that the reduced crossed product of the reduced cross-sectional algebra of a Fell bundle by the dual coaction is liminal, postliminal, or nuclear, if and only if so is the unit fiber of the bundle.

In the category of topological spaces with continuous maps, an action \(\beta \) on \(Y\) is an enveloping action of the partial action \(\alpha \) on \(X\), if \(X\) is an open subset of \(Y\), \(\alpha =\beta | _X\), and \(Y\) is the \(\beta \) -orbit of \(X\). He shows that in this category the enveloping action always exists and it is unique up to isomorphism.

In the category of \(C^{*}\)-algebras with their homomorphisms, an action \(\beta \) on \(B\) is an enveloping action of the partial action \(\alpha \) on \(A\), if \(A\) is a closed two-sided ideal of \(B\), \(\alpha =\beta | _A\), and \(B\) is the closed linear \(\beta \) -orbit of \(A\). The author shows that in this category the enveloping action is unique to up isomorphism if it exists, but in general it does not exist. For this reason, the author considers a weaker notion of enveloping action, called Morita enveloping action. To define the notion of Morita enveloping action, he introduces on the set of all partial actions a relation, called Morita equivalence, and studies its properties. An action \(\beta \) on \(B\) is a Morita enveloping action of the partial action \(\alpha \) on \(A\), if \(\beta \) is the enveloping action of a partial action that is Morita equivalent to \(\alpha \). He shows that any partial action has a Morita enveloping action, and this is unique up to Morita equivalence. Moreover, he proves a Takai duality theorem for partial actions: if \(\alpha \) is a partial action of an abelian group \(G\) on a \(C^{*}\)-algebra \(A\), \(\delta \) is the dual coaction of \(G\) on the reduced crossed product \(A\times _{\alpha ,r}G\), and \(\widehat{\delta }\) is the dual action of \(G\) on the reduced crossed product \(\left( A\times _{\alpha ,r}G\right) \times _{\delta ,r}\widehat{G}\) , then \(\widehat{\delta }\) is the Morita enveloping action of \(\alpha \).

As an application, it is shown that the reduced crossed product of the reduced cross-sectional algebra of a Fell bundle by the dual coaction is liminal, postliminal, or nuclear, if and only if so is the unit fiber of the bundle.

Reviewer: Maria Joita (Bucureşti)

### MSC:

46L05 | General theory of \(C^*\)-algebras |

46L55 | Noncommutative dynamical systems |

43A07 | Means on groups, semigroups, etc.; amenable groups |

### Keywords:

partial actions; enveloping actions; crossed product by partial actions; Takai duality; Fell bundles### References:

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