Partial actions of groups on algebras, a survey

C-crossed Products by Partial Actions and Actions of Inverse Semigroups

The recently developed theory of partial actions of discrete groups on C-algebras is extended. A related concept of actions of inverse semigroups on C-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.

Associativity of Crossed Products by Partial Actions, Enveloping Actions and Partial Representations

Given a partial action α 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 αG is associative, provided that A is semiprime. We also give a criterion for the existence of a global extension of ...

Characterisations of Crossed Products by Partial Actions

Partial actions of discrete groups on C∗-algebras and the associated crossed products have been studied by Exel and McClanahan. We characterise these crossed products in terms of the spectral subspaces of the dual coaction, generalising and simplifying a theorem of Exel for single partial automorphisms. We then use this characterisation to identify the Cuntz algebras and the Toeplitz algebras o...

D ec 1 99 7 PARTIAL DYNAMICAL SYSTEMS AND C ∗ - ALGEBRAS GENERATED BY PARTIAL ISOMETRIES

A collection of partial isometries whose range and initial projections satisfy a specified set of conditions often gives rise to a partial representation of a group. The Calgebra generated by the partial isometries is thus a quotient of the universal C-algebra for partial representations of the group, from which it inherits a crossed product structure, of an abelian C-algebra by a partial actio...

Reiter’s Properties for the Actions of Locally Compact Quantum Goups on von Neumann Algebras