C*-algebras in Kirillov Theory

Open Access
Author:
Bannangkoon, Pichkitti
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
August 13, 2015
Committee Members:
  • Nigel David Higson, Dissertation Advisor
  • John Roe, Committee Member
  • Nathanial Patrick Brown, Committee Member
  • Martin Bojowald, Committee Member
Keywords:
  • Kirillov theory
  • C*-algebras
  • unitary representations of nilpotent groups
  • regular representation
  • nilpotent Lie groups
Abstract:
In this dissertation, I study connections between C*-algebra theory and the representation theory of simply connected nilpotent Lie groups, specifically the Kirillov theory. If G is a connected, simply connected, nilpotent Lie group, then Kirillov’s famous theorem gives an explicit bijection between the set of equivalence classes of unitary irreducible representations of G and the set of coadjoint orbits of G in g*, the dual of the Lie algebra of G. Inspired by this, and by the Plancherel theorem, I introduce two new C*- algebras. The first is an algebra of operators on L2(G) and the second is an algebra of operators on L2(g*). I formulate the conjecture that they are isomorphic, prove the conjecture in the case of Heisenberg group (which is the crucial building block for general nilpotent Lie groups) and examine the prospects for the conjecture in other cases.