Asymptotically Contained Representations and the Spherical Plancherel Formula
Open Access
Author:
Tan, Qijun
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
November 30, 2018
Committee Members:
Nigel David Higson, Dissertation Advisor/Co-Advisor Nigel David Higson, Committee Chair/Co-Chair Paul Frank Baum, Committee Member Ping Xu, Committee Member Murat Gunaydin, Outside Member
Keywords:
Plancherel formula representation of semi-simple Lie groups symplectic geometry noncommutative geometry harmonic analysis C^*-algebra
Abstract:
We introduce the notion of asymptotic containment of representations of $C^*$-algebras. The spectral measure of the ambient representation is closely related to the spectral measure of an asymptotically contained one. When $G$ is a real reductive Lie group with Iwasawa decomposition $KAN$, and when $M$ is the center of $A$ in $K$, we show that the action of $C^*(G//K)$ on $L^2(K\backslash G/MN)$ is asymptotically contained in its action on $L^2(G//K)$. This fact can be used to prove Harish-Chandra's spherical Plancherel formula. Part of this thesis is a joint work with Nigel Higson.
