Noncommutative boundaries and the ideal structure of reduced crossed products

@article{Kennedy2019NoncommutativeBA,
title={Noncommutative boundaries and the ideal structure of reduced crossed products},
author={Matthew Kennedy and Christopher Schafhauser},
journal={Duke Mathematical Journal},
year={2019}
}
• Published 5 October 2017
• Mathematics
• Duke Mathematical Journal
A C*-dynamical system is said to have the ideal separation property if every ideal in the corresponding crossed product arises from an invariant ideal in the C*-algebra. In this paper we characterize this property for unital C*-dynamical systems over discrete groups. To every C*-dynamical system we associate a "twisted" partial C*-dynamical system that encodes much of the structure of the action. This system can often be "untwisted," for example when the algebra is commutative, or when the…
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References

SHOWING 1-10 OF 51 REFERENCES
The ideal structure of reduced crossed products
Let (A, G) be a C*-dynamical system with G discrete. In this paper we in- vestigate the ideal structure of the reduced crossed product C*-algebra and in particular we determine sufficient—and in some
Reduced twisted crossed products over C*-simple groups
• Mathematics
• 2016
We consider reduced crossed products of twisted C*-dynamical systems over C*-simple groups. We prove there is a bijective correspondence between maximal ideals of the reduced crossed product and
Topologically free actions and ideals in discrete C*-dynamical systems
• Mathematics
Proceedings of the Edinburgh Mathematical Society
• 1994
A C*-dynamical system is called topologically free if the action satisfies a certain natural condition weaker than freeness. It is shown that if a discrete system is topologically free then the ideal
Uniformly recurrent subgroups and the ideal structure of reduced crossed products
We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a
Some Simple C*-Algebras Constructed As Crossed Products with Discrete Outer Automorphism Groups
An analogue for C*-algebras is given of the theorem of von Neumann (Theorem VIII of [24]) that the crossed product of a commutative von Neumann algebra by a discrete group acting freely and
Injective envelopes of dynamical systems
By dynamical systems in the title we mean the objects, called G-modules, of a category CG consisting of operator spaces with a certain L(G)-module structure and the complete contractions commuting
C*-simplicity and the unique trace property for discrete groups
• Mathematics
• 2014
A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical
Boundaries of reduced C*-algebras of discrete groups
• Mathematics
• 2014
For a discrete group G, we consider the minimal C*-subalgebra of $\ell^\infty(G)$ that arises as the image of a unital positive G-equivariant projection. This algebra always exists and is unique up
STRUCTURE FOR REGULAR INCLUSIONS. I
We study pairs (C,D) of unital C*-algebras where D is a regular abelian C*-subalgebra of C. When D is a MASA in C, we prove the existence and uniqueness of a completely positive unital map E of C