C*-algebras associated to group actions on trees, and connections to classification
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Open Access
Type
ThesisThesis type
Doctor of PhilosophyAuthor/s
Wu, VictorAbstract
We study C*-algebras arising from actions of countable, discrete groups on directed trees. Such an
action gives rise to a C*-algebra in two ways: the action induces an action of the group on the
boundary of the tree, from which one can construct a crossed product C*-algebra; but ...
See moreWe study C*-algebras arising from actions of countable, discrete groups on directed trees. Such an action gives rise to a C*-algebra in two ways: the action induces an action of the group on the boundary of the tree, from which one can construct a crossed product C*-algebra; but also the action gives rise to a (directed) graph of groups as a quotient object, and one can also associate a C*- algebra to this directed graph of groups. After exploring these two C*-algebras as part of separate, more general classes (crossed products arising from group actions on the boundaries of multitrees, and C*-algebras of groupoid-embeddable categories), we show that the crossed product C*-algebra is Morita equivalent to the directed graph-of-groups C*-algebra. Finally, we show that directed graphof- groups C*-algebras (and their corresponding crossed products) contain all stable UCT Kirchberg algebras (a class of C*-algebras classified completely by K-theory).
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See moreWe study C*-algebras arising from actions of countable, discrete groups on directed trees. Such an action gives rise to a C*-algebra in two ways: the action induces an action of the group on the boundary of the tree, from which one can construct a crossed product C*-algebra; but also the action gives rise to a (directed) graph of groups as a quotient object, and one can also associate a C*- algebra to this directed graph of groups. After exploring these two C*-algebras as part of separate, more general classes (crossed products arising from group actions on the boundaries of multitrees, and C*-algebras of groupoid-embeddable categories), we show that the crossed product C*-algebra is Morita equivalent to the directed graph-of-groups C*-algebra. Finally, we show that directed graphof- groups C*-algebras (and their corresponding crossed products) contain all stable UCT Kirchberg algebras (a class of C*-algebras classified completely by K-theory).
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Date
2024Rights statement
The author retains copyright of this thesis. It may only be used for the purposes of research and study. It must not be used for any other purposes and may not be transmitted or shared with others without prior permission.Faculty/School
Faculty of Science, School of Mathematics and StatisticsAwarding institution
The University of SydneyShare