Abstract. Given a partial action α of a group G on an associative algebra A, we consider the crossed product A α G. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of A α G obtained in the context of C * -algebras. In particular, we prove that A α G is associative, provided that A is semiprime. We also give a criterion for the existence of a global extension of a given partial action on an algebra, and use crossed products to study relations between partial actions of groups on algebras and partial representations. As an application we endow partial group algebras with a crossed product structure.

In this article, among other results, we develop a Galois theory of commutative rings under partial actions of finite groups, extending the well-known results by In the celebrated paper by Chase, Harrison and Rosenberg [3] the authors developed a Galois theory for commutative ring extensions S ⊃ R, under the assumptions that S is separable over R, finitely generated and projective as an R-module, and the elements of the Galois group G are pairwise strongly distinct R-automorphisms of S. In particular, Theorem 1.3 of that paper gives several equivalent conditions for the definition of a Galois extension and Theorem 2.3 states a one-to-one correspondence between the subgroups of G and the R-subalgebras of S which are separable and G-strong.On the other hand, partial actions of groups have been introduced in the theory of operator algebras giving powerful tools of their study (see, in particular, [6,7,10,16] and [18]). A related concept, that of a partial representation of a group on a Hilbert space, has been defined independently by Exel [7], and Quigg and Raeburn [18]. Several relevant classes of C * -algebras were deeply investigated in [8-10] from the point of view of partial actions and partial representations of groups, including the Cuntz-Krieger algebras introduced in [4].Given a partial action of a group on an object it is natural to ask whether it is a restriction of a global action defined on a bigger object. Such global action is called a globalization or an enveloping action, provided that certain minimality condition is satisfied which guarantees its uniqueness. Globalizations of partial actions where first considered by F. Abadie in his PhD Thesis of 1999 (see also [1]) and independently by Kellendonk and Lawson in [15]. $ This paper was partially supported by CNPq, CAPES, FAPESP and FAPERGS (Brazil). (M. Dokuchaev), mferrero@mat.ufrgs.br (M. Ferrero), paques@ime.unicamp.br (A. Paques).

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For a twisted partial action Θ of a group G on an (associative nonnecessarily unital) algebra A over a commutative unital ring k, the crossed product A Θ G is proved to be associative. Given a G-graded k-algebra B = g∈G B g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B 1 Θ G for some twisted partial action of G on B 1 . The equalityis one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G.

We study the relations between partial and global group cohomology. We show, in particular, that given a unital partial action of a group G on a ring A, such that A is a direct product of indecomposable rings, then any partial n-cocycle with values in A is globalizable.