Fri, 30 Oct 2026
14:00
14:00
L5
Supports of Ideals in Reduced Groupoid C*-Algebras and Applications
Kang Li
(Lund University)
Abstract
Based on previous results on ideals in reduced groupoid C*-algebras, Christian Bönicke and I proved in 2018 that all ideals in a reduced groupoid C*-algebra C_r*(G) are dynamical if and only if the underlying étale groupoid G is inner exact and has the residual intersection property. Recall that an ideal I in the reduced groupoid C*-algebra C_r*(G) is dynamical if I is uniquely determined by an open invariant subset U of G^0, the unit space of the groupoid G. Equivalently, I=I(U) the ideal generated by C_0(U) in C_r*(G).
In the joint work with Jiawen Zhang, we are investigating ideals in the reduced C*-algebra of a general étale groupoid G. Generally, each ideal I in C_r*(G) is naturally associated with two open invariant subsets of G^0, called the inner support U_I and the outer support V_I for the given ideal I. By introducing the notion of ghostly ideals in C_r*(G), we are able to extend the sandwiching result for ideals in C_r*(G) from inner exact groupoids to general étale groupoids. Moreover, we use effectiveness and the intersection property to characterize when the inner and outer supports of a given ideal in C_r*(G) coincide. If time permits, I will explain several applications to regular ideals, tracial ideals and amenability problem for the Thompson's group F.