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

By Mikhailo Dokuchaev, R. Exel. 2002

265 citations recorded by the source corpus.

Given a partial action a 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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/325ce574b49cd7de17469ac955e6ae5def64f660

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Partial Groupoid Actions: Globalization, Morita Theory, and Galois Theory (2011, background): “A purely algebraic study of partial group actions was done in [12] and a corresponding Galois theory was developed in [13], stimulating further investigations by several others authors, see [11] for a more extensive bibliography.”
Leavitt Path Algebras as Partial Skew Group Rings (2012, background): “It is our intention to connect the theory of Leavitt path algebras with another key concept arising from operator algebras theory, namely, partial skew group rings (or just skew group rings, depending on the literature, see [6, 10, 15]).”
Globalization theorems for partial Hopf (co)actions, and some of their applications (2010, background): “This is the globalization theorem as one can find in [7].”
Recent developments around partial actions (2018, background): “The first algebraic results on the subject appeared in [151], [120], [204], [275], [91], [276], [205], [89] and [117], 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”
Partial actions and Galois theory (2007, background): “Let S be an α-partial Galois extension of R and T be a commutative R-algebra.”
Enveloping Actions for Partial Hopf Actions (2008, methodology): “The developments originated by the definition of partial group actions include crossed products [13], partial representations [9, 5] and soon this theme became an independent topic of interest in ring theory [6, 10].”
Partial Dynamical Systems, Fell Bundles and Applications (2015, background): “I eventually managed to attract the attention of Misha Dokuchaev, and together we were able to understand exactly what was going on [32].”