Associativity of crossed products by partial actions, enveloping actions and partial representations

Abstract

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.

Recent developments around partial actions

“…generated by an idempotent which is central in A. These are exactly those partial actions of G on a unital ring A which are globalizable (see [117]). Moreover, if each domain is an s-unital ring, then α is called an s-unital partial action on A (see [8,42]).…”

“…The first algebraic results on the subject appeared in [89,91,117,120,151,204,205,275,276], including the first algebraic application for tiling semigroups in [204] (with further use in [283]), for E-unitary inverse semigroups in [205], to inverse semigroups and F-inverse monoids in [275] and to inverse monoids of Möbius type in [91]. Independently from Exel's definition of a partial action, Coulbois [110] used a more restrictive notion, called a pre-action, to deal with the Ribes-Zalesski property (R Z n ) of groups from model theoretic point of view (see also [111]).…”

“…Since then the algebraic approach is being developed in diverse directions in various levels of generality, including partial actions of Hopf (or, more generally, weak Hopf) algebras [15,[17][18][19][20][21][22]48,69,72,[78][79][80][81][82][83][86][87][88]165,250,282], semigroups [68,97,132,[197][198][199]209,213,220,227,233,234,242,255], inductive constellations [198], groupoids [37,[40][41][42][43]178], and, more generally, categories [244]. In particular, further algebraic applications have been found to graded algebras [117,121], to Hecke algebras [153], to Leavitt path algebras [184,…”

“…2 This was studied first in the PhD Thesis [3] (see also [4]) and, independently from [3,4], in [205,276]. Subsequent results were obtained in [27,38,102,117,122,136,156,169], and more recently in [8,50,51,105,168,262,263]. The question was also considered for partial semigroup actions in [197,199,209,213,227,233], for partial groupoid actions in [41,42,178], and around partial Hopf (co)actions in [15,18,19,21,[78][79][80].…”

“…The regularity assumption imposes a mild restriction on the domains of the partial isomorphisms involved in a partial action, more precisely, the intersection of domains are assumed to coincide with their product. This is a suitable constraint since, on one hand, it resolves the discrepancy between the definitions of partial actions given in [117,121], and on the other, in almost all investigations on the subject the considered partial group actions on algebras (or rings) are regular, so that this concept provides a sufficiently general framework for the theory.…”