TY - BOOK

T1 - Boundaries, injective envelopes, and reduced crossed products

AU - Bryder, Rasmus Sylvester

PY - 2017

Y1 - 2017

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.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122894616105763

M3 - Ph.D. thesis

BT - Boundaries, injective envelopes, and reduced crossed products

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -