Kang Li (Erlangen): Type decomposition of ideals in reduced groupoid C*-algebras. Oberseminar C*-Algebren.
Tuesday, 29.10.2024 16:15 im Raum SRZ 216/217
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 ongoing project with Jiawen Zhang, we are investigating ideals in reduced C*-algebras of general étale groupoids. In this general setting, each ideal I in C_r*(G) is associated with two open invariant subsets of G^0, namely the inner support U_I and the outer support V_I. Let I be an ideal in C_r*(G), we call I is of type I if I(U_I ) ? I ? I(V_I ), and I is of type II if U_I = V_I. It is known that an ideal I in C_r^*(G) is dynamical if and only if I is of both type I and type II; and a non-dynamical ideal I can be either type I or type II. It turns out that every ideal in C_r*(G) can be uniquely reconstructed from type I and type II ideals. Most importantly, we also provide characterizations for type I and type II ideals by means of underlying groupoids.
In my talk I will present several examples and applications.
Angelegt am 10.10.2024 von Elke Enning
Geändert am 10.10.2024 von Elke Enning
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