Associativity of crossed products by partial actions, enveloping actions and partial representations. (English) Zbl 1072.16025

Dokuchaev, M.; Exel, R.

Trans. Am. Math. Soc. 357, No. 5, 1931-1952 (2005).

Let \(G\) be a group and \(\mathcal A\) an associative not necessarily unital algebra over the field \(K\). A partial action \(\alpha\) of \(G\) on \(\mathcal A\) is a collection of ideals \({\mathcal D}_g\) of \(\mathcal A\) and algebra isomorphisms \(\alpha_g\colon{\mathcal D}_{g^{-1}}\to{\mathcal D}_g\) such that: (i) \({\mathcal D}_1=\mathcal A\) and \(\alpha_1\) is the identity map; (ii) \({\mathcal D}_{(gh)^{-1}}\supseteq\alpha_h^{-1}({\mathcal D}_h\cap{\mathcal D}_{g^{-1}})\); (iii) \(\alpha_g\circ\alpha_h(x)=\alpha_{gh}(x)\) for each \(x\in\alpha_h^{-1}({\mathcal D}_h\cap{\mathcal D}_{g^{-1}})\). If \(\alpha\) is given, the crossed product \({\mathcal A}\rtimes_\alpha G\) is the set of finite formal sums \(\{\sum_{g\in G}a_g\delta_g\mid a_g\in{\mathcal D}_g\}\).
These concepts, and the results of this paper are motivated by, and generalize results from the theory of operator algebras. The authors give necessary conditions for the associativity of \({\mathcal A}\rtimes_\alpha G\), and show that \({\mathcal A}\rtimes_\alpha G\) is not associative in general, but it is associative provided that \(\mathcal A\) is semiprime.
Another main result gives necessary and sufficient conditions for the existence of a so called global extension (also called enveloping action) for a partial action on a unital algebra. Finally, crossed products are used to relate partial actions with partial representations, and several applications are given.

Reviewer: Andrei Marcus (Cluj-Napoca)

MSC:

16S99	Associative rings and algebras arising under various constructions
16S35	Twisted and skew group rings, crossed products
16S10	Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16S34	Group rings
20C07	Group rings of infinite groups and their modules (group-theoretic aspects)
20C15	Ordinary representations and characters

Keywords:

partial actions; crossed products; partial representations; partial group rings; gradings; groupoids

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