On partial actions and groupoids

Abstract

Abstract. We prove that, as in the case of global actions, any partial action gives rise to a groupoid provided with a Haar system, whose C * -algebra agrees with the crossed product by the partial action.

Recent developments around partial actions

“…The importance of groupoids for partial representations was mentioned above. On the other hand, in [5] a locally compact groupoid (called transformation groupoid) was related to any continuous partial action θ of a second countable locally compact Hausdorff group G on a second countable locally compact Hausdorff space X and used to prove that the C * -algebra of the groupoid is isomorphic to the C * -crossed product C 0 (X ) full θ G (here the partial action of G on C 0 (X ), dual to that of G on X, is denoted by the same symbol θ ). In the discrete group case another way to associate a groupoid to a partial action was given in [6].…”

“…As it is was mentioned in many occasions, the formal concept of a partial action (in the sense we are using it) appeared in the theory C * -algebras (see [146,149,151,232]), permitting one to endow relevant classes of C * -algebras with a general structure of a partial crossed product [147,148,150,160,264] (see also [155]), and promptly stimulating further use and discussions in the area [1,3,4,150,152,157,158,264,[271][272][273]. Subsequent C * -algebraic and topological developments on partial actions were made in [5,6,11,60,93,112,141,159,163,222,233].…”

“…The latter stimulated also further Galois theoretic results in [69], which were based on a coring C constructed for a unital partial action of a finite group, offering thus a more conceptual approach to partial Galois theory via Galois corings. 5 The coring C was shown to fit the general theory of cleft bicomodules in [65], and, in addition, in [66] descent theory for corings was applied, using C, to define non-Abelian Galois cohomology (i = 0, 1) for unital partial Galois actions of finite groups. The article by Caenepeel and Janssen [72], in its turn, became the starting point for a series of investigation of partial Hopf (co)actions (see the articles cited above), in particular, several globalization results were obtained in [15,18,19,[78][79][80]212].…”

“…Then for any σ ∈ Z 2 (G, I ; A) it is possible to construct a central extension A → E → M, a partial homomorphism ψ : G → M and its lifting (section) ϕ : G → E, such that the diagram (7) is commutative upon the replacement of κ * by A, and σ plays the role of the factor set related to ϕ as in (3), (4), (5). This leads to a bijection between the elements of H 2 (G, I ; A) and the equivalence classes of certain appropriately defined central extensions [137,Theorem 3.3].…”

“…when char κ divides |G 1 |) was established in [135,Theorem 3.7]. The above mentioned counter-example from [120] for the isomorphism question was improved in [138] by giving counter-examples over C for orders |G| = 2 5 and |G| = 3 5 , and pointing out that there are no counter-examples for |G| < 2 5 and |G| = p n < 3 5 , p = 2.…”