Boundaries, injective envelopes, and reduced crossed products

Rasmus Sylvester Bryder

Boundaries, injective envelopes, and reduced crossed products

Rasmus Sylvester Bryder

Abstract

In this dissertation, we study boundary actions, equivariant injective envelopes, as well as theideal structure of reduced crossed products. These topics have recently been linked to thestudy of C-simple groups, that is, groups with simple reduced group C-algebras.In joint work with Matthew Kennedy, we consider reduced twisted crossed products overC-simple groups. For any twisted C-dynamical system over a C-simple group, we provethat there is a one-to-one correspondence between maximal invariant ideals in the underlyingC-algebra and maximal ideals in the reduced crossed product. When the amenable radical ofthe underlying group is trivial, we verify a one-to-one correspondence between invariant tracialstates on the underlying C-algebra and tracial states on the reduced crossed product.In subsequent joint work with Tron Omland, we give criteria ensuring C-simplicity and theunique trace property for a non-ascending countable HNN extension. This is done by bothpurely algebraic and dynamical methods. Moreover, we also characterize C-simplicity of anHNN extension in terms of its boundary action on its Bass-Serre tree.We finally consider equivariant injective envelopes of unital C*-algebras, and relate the intersection property for group actions on unital C*-algebras to the intersection property for theequivariant injective envelope. Moreover, we also prove that the equivariant injective envelopeof the centre of the injective envelope of a unital C*-algebra can be regarded as a C*-subalgebraof the centre of the equivariant injective envelope of the original C*-algebra.

Originalsprog	Engelsk

Forlag	Department of Mathematical Sciences, Faculty of Science, University of Copenhagen
Status	Udgivet - 2017

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title = "Boundaries, injective envelopes, and reduced crossed products",

abstract = "In this dissertation, we study boundary actions, equivariant injective envelopes, as well as theideal structure of reduced crossed products. These topics have recently been linked to thestudy of C-simple groups, that is, groups with simple reduced group C-algebras.In joint work with Matthew Kennedy, we consider reduced twisted crossed products overC-simple groups. For any twisted C-dynamical system over a C-simple group, we provethat there is a one-to-one correspondence between maximal invariant ideals in the underlyingC-algebra and maximal ideals in the reduced crossed product. When the amenable radical ofthe underlying group is trivial, we verify a one-to-one correspondence between invariant tracialstates on the underlying C-algebra and tracial states on the reduced crossed product.In subsequent joint work with Tron Omland, we give criteria ensuring C-simplicity and theunique trace property for a non-ascending countable HNN extension. This is done by bothpurely algebraic and dynamical methods. Moreover, we also characterize C-simplicity of anHNN extension in terms of its boundary action on its Bass-Serre tree.We finally consider equivariant injective envelopes of unital C*-algebras, and relate the intersection property for group actions on unital C*-algebras to the intersection property for theequivariant injective envelope. Moreover, we also prove that the equivariant injective envelopeof the centre of the injective envelope of a unital C*-algebra can be regarded as a C*-subalgebraof the centre of the equivariant injective envelope of the original C*-algebra.",

author = "Bryder, {Rasmus Sylvester}",

year = "2017",

language = "English",

publisher = "Department of Mathematical Sciences, Faculty of Science, University of Copenhagen",

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N2 - In this dissertation, we study boundary actions, equivariant injective envelopes, as well as theideal structure of reduced crossed products. These topics have recently been linked to thestudy of C-simple groups, that is, groups with simple reduced group C-algebras.In joint work with Matthew Kennedy, we consider reduced twisted crossed products overC-simple groups. For any twisted C-dynamical system over a C-simple group, we provethat there is a one-to-one correspondence between maximal invariant ideals in the underlyingC-algebra and maximal ideals in the reduced crossed product. When the amenable radical ofthe underlying group is trivial, we verify a one-to-one correspondence between invariant tracialstates on the underlying C-algebra and tracial states on the reduced crossed product.In subsequent joint work with Tron Omland, we give criteria ensuring C-simplicity and theunique trace property for a non-ascending countable HNN extension. This is done by bothpurely algebraic and dynamical methods. Moreover, we also characterize C-simplicity of anHNN extension in terms of its boundary action on its Bass-Serre tree.We finally consider equivariant injective envelopes of unital C*-algebras, and relate the intersection property for group actions on unital C*-algebras to the intersection property for theequivariant injective envelope. Moreover, we also prove that the equivariant injective envelopeof the centre of the injective envelope of a unital C*-algebra can be regarded as a C*-subalgebraof the centre of the equivariant injective envelope of the original C*-algebra.

AB - In this dissertation, we study boundary actions, equivariant injective envelopes, as well as theideal structure of reduced crossed products. These topics have recently been linked to thestudy of C-simple groups, that is, groups with simple reduced group C-algebras.In joint work with Matthew Kennedy, we consider reduced twisted crossed products overC-simple groups. For any twisted C-dynamical system over a C-simple group, we provethat there is a one-to-one correspondence between maximal invariant ideals in the underlyingC-algebra and maximal ideals in the reduced crossed product. When the amenable radical ofthe underlying group is trivial, we verify a one-to-one correspondence between invariant tracialstates on the underlying C-algebra and tracial states on the reduced crossed product.In subsequent joint work with Tron Omland, we give criteria ensuring C-simplicity and theunique trace property for a non-ascending countable HNN extension. This is done by bothpurely algebraic and dynamical methods. Moreover, we also characterize C-simplicity of anHNN extension in terms of its boundary action on its Bass-Serre tree.We finally consider equivariant injective envelopes of unital C*-algebras, and relate the intersection property for group actions on unital C*-algebras to the intersection property for theequivariant injective envelope. Moreover, we also prove that the equivariant injective envelopeof the centre of the injective envelope of a unital C*-algebra can be regarded as a C*-subalgebraof the centre of the equivariant injective envelope of the original C*-algebra.

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