Enveloping actions and Takai duality for partial actions

By F. Abadie. 2000

143 citations recorded by the source corpus.

We show that any partial action on a topological space X is the restriction of a suitable global action, called enveloping action, that is essentially unique. In the case of C∗-algebras, we prove that any partial action has a unique enveloping action up to Morita equivalence, and that the corresponding reduced crossed products are Morita equivalent. The study of the enveloping action up to Morita equivalence reveals the form that Takai duality takes for partial actions. By applying our constructions, we prove that the reduced crossed product of the reduced cross-sectional algebra of a Fell bundle by the dual coaction is liminal, postliminal, or nuclear, if and only if so is the unit fiber of the bundle. We also give a non-commutative generalization of the well-known fact that the integral curves of a vector field on a compact manifold are defined on all of R.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b8734ca543d6460e3a2baadf4135f5874a62a316

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Citations in the corpus

Partial actions and Galois theory (2007, methodology): “Abadie in his PhD Thesis of 1999 (see also [1]) and independently by Kellendonk and Lawson in [15].”
Partial Groupoid Actions: Globalization, Morita Theory, and Galois Theory (2011, background): “This question was first considered by Abadie [1] in the context of continuous partial group actions on topological spaces and C -algebras.”
Dynamical systems associated to separated graphs, graph algebras, and paradoxical decompositions☆ (2012, methodology): “We use globalization results from [1] to reach the desired goal with global actions.”
Enveloping Actions for Partial Hopf Actions (2008, methodology): “Thus we may ask, which conditions on a partial action enable us to say that this partial action is a restriction of a global action? The first result concerning enveloping actions was proved in the context of C∗ algebras in [1]; to this intent, the author used techniques of Fell Bundles and Hilbert ”
On partial actions and groupoids (2004, background): “In the first section we define partial actions on topological spaces (see also [1]), and we show that our definition is the dual notion of partial actions on C∗-algebras.”
Recent developments around partial actions (2018, methodology): “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], [232], [149], [151]), permitting one to endow relevant classes of Calgebras with a general structure of a partial crossed product [147], [148]”
ON INVERSE SEMIGROUP C ∗ -ALGEBRAS AND CROSSED PRODUCTS (2011, methodology): “Khoshkam and Skandalis prove a more general result than that of Abadie in [12] using different terminology.”
Globalization of confluent partial actions on topological and metric spaces (2002, unknown): “ the map {x}→{y}by (yx) for x6= yin X. Then we have a subspace Zof Y formed by all points of the form xor (y0) ·x. The space Zis homeomorphic to the quotient space obtained by taking one base copy of [0,1] Globalization of Conﬂuent Partial Actions 27 and uncountably many copies of [0,1] indexed over”
Partial actions: a survey (2011, background): “Abadie in the context of continuous partial group actions on topological spaces (in particular, on abstract sets) and C∗-algebras in his PhD Thesis of 1999 [1] (see also [2]), and independently by J.”
Associativity of crossed products by partial actions, enveloping actions and partial representations (2002, background): “The following result is inspired by F. Abadie’s thesis [1] (see also [ 2 ]):”
Partial Dynamical Systems, Fell Bundles and Applications (2015, methodology): “Using (15.3) (which is stated for Hilbert modules, and hence also holds for C*-algebras) we then have x = lim x(x∗x)1/n ∈ 〈J ∩B1〉, We therefore conclude that J ∩Bg ⊆ 〈J ∩ B1〉, from where it follows that K ⊆ 〈J ∩B1〉, as desired.”
Crossed products by twisted partial actions and graded algebras (2008, background): “Interesting results have been obtained also in [36] and [38] (with respect to the topological part of [36] see also [1]).”