Unitary groups of C*-algebras
Abstract. Given a unital C*-algebra we consider two associated topological groups
(that are Polish if A is separable): The connected component of the unit
of the unitary group of A, denoted by U_A, and the group of
approximately inner automorphisms of A, denoted by V_A, which is the
closure of the image of U_A in Aut(A) via the map u-> Ad(u). In my talk
I will show how using basic Lie theory, refining the previous work of
Leonel Robert, we can quite well relate the closed normal subgroup
structures of U_A and V_A with the ideal structure of A.
Sample results that will be mentioned during the talk:
(1) If A is a locally AF algebra then there is a one-to-one
correspondence between closed normal subgroups of V_A and ideals of A
that do not admit a character; in the separable case, the closed normal
subgroups of V_A can be then conveniently read off of a Bratteli diagram
of A.
(2) If A is a reduced group C*-algebra of a countable discrete
non-amenable group, then A is simple if and only if U_A/T is
topologically simple, where T is the canonical torus inside A.
I won't assume any particular knowledge of C*-algebras or Lie theory.
Based on recent joint work with Hiroshi Ando.
