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• W2739672401 abstract "We give a short survey on partial actions, partial representations and related notions. In the theory of C ∗ -algebras a concept of a partial action was introduced as an efficient tool of their study, permitting to characterize various important classes of C ∗ -algebras as crossed products by partial actions. Such a characterization made it possible to obtain relevant results on K-theory, ideal structure and representa- tions of the algebras under consideration. Also amenability questions, especially amenability of C ∗ -algebraic bundles (also called Fell bundles) 1 , were successfully investigated using partial actions and the related concept of a partial representa- tion. Crossed products play a central role in the rich interaction between operator algebras and dynamical systems, and partial actions on C ∗ -algebras (= partial C ∗ - dynamical systems) provide an appropriate approach to C ∗ -algebras generated by partial isometries. Amongst prominent classes of C ∗ -algebras endowed with the structure of non-trivial crossed products by partial actions one may list the Bunce- Deddens and the Bunce-Deddens-Toeplitz algebras (42), the approximately finite dimensional algebras (43), the Toeplitz algebras of quasi-ordered groups, as well as the Cuntz-Krieger algebras (52), (88). The notion of a partial action on a C ∗ -algebra appeared first time in the li- terature in Ruy Exel's paper (41 )i n whichC ∗ -algebraic crossed products by a partial automorphism (equivalently, by a partial action of the infinite cyclic group) were introduced and studied from the point of view of their internal structure, K- theory and representations. Motivated by a dynamical system point of view, Exel's main purpose was to develop a method which allows to describe the structure of C ∗ - algebras possessing actions of the circle group. The possibility for a straightforward generalization of the main construction was also noted. Exel's paper was followed by K. McClanahan's article (74) in which the formal definition of a C ∗ -crossed pro- duct by a partial action of a discrete group was given, permitting to obtain further K-theoretic results. The general notion of a (continuous) twisted partial action of" @default.
• W2739672401 created "2017-08-08" @default.
• W2739672401 creator A5011535782 @default.
• W2739672401 date "2011-01-01" @default.
• W2739672401 modified "2023-09-25" @default.
• W2739672401 title "Partial actions: a survey" @default.
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• W2739672401 doi "https://doi.org/10.1090/conm/537/10573" @default.
• W2739672401 hasPublicationYear "2011" @default.
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