Construction of globalizations for partial actions on rings, algebras, C$^*$-algebras and Hilbert bimodules

Recent developments around partial actions

“…More recent C * and topological advances include the groupoid approach to the enveloping C * -algebras associated to partial actions of countable discrete groups on (locally) compact spaces in [156], the use of inverse semigroup expansions to treat C *crossed products by twisted partial actions via twisted global actions of the expansion in [68], the full (respectively, reduced) partial C * -crossed product descriptions of full (respectively, reduced) C * -algebras of countable E-unitary or strongly 0-E-unitary inverse semigroups as well as of tight groupoids of countable strongly 0-E-unitary inverse semigroups in [235], the study of the continuous orbit equivalence for partial dynamical systems and of the partial transformation groupoids with applications to graph C * -algebras and semigroup C * -algebras in [224], the partial group action approach to produce a Bratteli-Vershik model linked to a minimal homeomorphism between open subsets with finite disjoint complements of the Cantor set in [179], new developments on the globalization problem for partial actions on C * -algebras and Hilbert bimodules in [168], the employment of the partial crossed product theory to the investigation of the Cuntz-Li C * -algebras related to an integral domain in [61] with a further development in [281] for C * -algebras associated with an injective endomorphism of a group with finite cokernel. Moreover, partial crossed products turned out to be useful to deal with C * -algebras arising from self-similar graph actions in [162], with C * -algebras associated to any stationary ordered Bratteli diagram in [185], as well as with ultragraph C * -algebras and related infinite alphabet shifts in [187][188][189]191].…”

“…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].…”

“…In addition, it is shown in [7,Corollary 4.3] (see also [10,Proposition 4.13]) that weakly equivalent Fell bundles have (strongly) Morita equivalent full and reduced cross-sectional C * -algebras. Notice that [217,Proposition 7.1] implies that the reduced C * -algebra of a Fell bundle over a discrete group G is Morita equivalent to the reduced crossed product by a global action of G. 4 In the recent article [168] the problem of the existence of an enveloping action for a partial group action on a non-necessarily unital (abstract) ring was investigated and applied to the globalization problem for partial actions on C * -algebras and equivalence Hilbert bimodules. More specifically, amongst various facts it was proved that if a partial action α = {α g : A g −1 → A g , g ∈ G} of a (discrete) group on a (non-necessarily unital) ring A admits a globalization, then the following condition is satisfied:…”

“…for all c ∈ A g (see [168,Theorem 2]). Under appropriate non-degeneracy assumptions (which always hold for C * -algebras) condition (2) becomes sufficient for the existence of a globalization for α.…”

“…Write G dis regarding G as a discrete group and let α dis be the C * -partial action of G dis on A given by α. Then [168,Corollary 2] means that for the globalization problem the topology of G can be forgotten, i.e. α admits a C * -globalization if and only of so does α dis .…”