Partial Actions of Groups on Cell Complexes

Partial actions of groups on generalized matrix rings

We show that for a partial action η with closed domain of a compact group G on a profinite space X the space of orbits X/~G is profinite, this leads to the fact that when G is profinite the enveloping space XG is also profinite. Moreover, we provide conditions for the induced quotient map πG : X → X / ∼G of η to have a continuous section. Relations between continuous sections of πG and continuous sections of the quotient map induced by the enveloping action of η are also considered. At the end of this work, we prove that the category of actions on profinite spaces with countable number of clopen sets is reflective in the category of actions of compact Hausdorff spaces having countable number of clopen sets.

In this survey paper we define partial actions of groups on algebras and give several related results. For any partial action a partial skew group ring is defined. This partial skew group ring is not always associative. Conditions under which associativity holds are studied. Several other questions are considered like, for example, enveloping actions and properties of partial skew group rings.

Actions of inverse semigroups arising from partial actions of groups

Nuclear dimension of crossed products associated to partial dynamical systems

J. Kellendonk and M. V. Lawson established that each partial action of a group G on a set Y can be extended to a global action of G on a set Y G containing a copy of Y. In this paper we classify transitive partial group actions. When G is a topological group acting on a topological space Y partially and transitively we give a condition for having a Hausdorff topology on Y G such that the global group action of G on Y G is continuous and the injection Y into Y G is an open dense equivariant embedding.

In this paper, we generalize in the context of partial group action a result of Sugano by giving some equivalent conditions for partial Galois extensions to be Hirata-separable extensions. As an application, we show that every partial Galois extension with inner partial group action is Hirata-separable.

Using an algebraic point of view we present an introduction to the groupoid theory; that is, we give fundamental properties of groupoids as uniqueness of inverses and properties of the identities and study subgroupoids, wide subgroupoids, and normal subgroupoids. We also present the isomorphism theorems for groupoids and their applications and obtain the corresponding version of the Zassenhaus Lemma and the Jordan-Hölder theorem for groupoids. Finally, inspired by the Ehresmann-Schein-Nambooripad theorem we improve a result of R. Exel concerning a one-to-one correspondence between partial actions of groups and actions of inverse semigroups.

Partial actions of groups on semiprime rings

We consider a large class of partial actions of groups on rings, called regular, which contains all s-unital partial actions, as well as all partial actions on C∗-algebras. For them the notion of Morita equivalence is introduced, and it is shown that any regular partial action is Morita equivalent to a globalizable one, and the globalization is essentially unique. It is also proved that Morita equivalent s-unital partial actions on rings with orthogonal local units are stably isomorphic. In addition, we show that Morita equivalent s-unital partial actions on commutative rings must be isomorphic, and an analogous result for C∗-algebras is also established. Date: November 8, 2015. This work was partially supported by CNPq of Brazil (Proc. 305975/2013-7, Proc. 300362/20102), Fapesp of Brazil (Proc. Proc. 2009/52665-0), MINECO (Ministerio de Economia y Competitividad), (Fondo Europeo de Desarrollo Regional) project MTM2012-35240, Spain, and Fundacion Seneca of Murcia, Programa Hispano Brasileno de Cooperacion Universitaria PHB2012-0135, Spain. 2000 Mathematics Subject Classification: Primary 16S35; Secondary 16W22, 46L05.

In this paper, we introduce the concept of partial groupoid actions on [Formula: see text]-semicategories as well as we give criteria for existence of a globalization of it. This point of view is a generalization of the notions of partial groupoid actions on rings and partial group action on an [Formula: see text]-semicategory. We also define the notions of partial skew groupoid category, smash product and describe functorial relations between them, in particular we show that the smash product is a Galois covering of its associated skew groupoid category.

Partial projective representations and partial actions II

We consider the category of partial actions, where the group and the set upon which the group acts can vary. Within this framework, we develop a theory of quotient partial actions and prove that this category is both (co)complete and encompasses the category of groupoids as a full subcategory. In particular, we establish the existence of a pair of adjoint functors, denoted as Φ : Grpd → PA and Ψ : PA → Grpd , with the property that Ψ Φ ≅ 1 Grpd . Next, for a given groupoid Γ, we provide a characterization of all partial actions that allow the recovery of the groupoid Γ through Ψ . This characterization is expressed in terms of certain normal subgroups of a universal group constructed from Γ .

Classification of free actions of finite groups on the 3-torus

We extend the classical notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A *α G is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.