AG Lie-Gruppen: X. Fu (Univ. Toronto): Matrix techniques and approximation by invertible

Matrix techniques and approximation by invertible

Vortragender: Xuanlong Fu, Univ. Toronto
Einladender: Kang Li

Abstract:

If every element in a unital C*-algebra can be approximated in norm by
invertible elements, then such C*-algebra is called has stable rank one. A
non-unital C*-algebra is called has stable rank one if its minimal unitization
has stable rank one. Stable rank one is a frequently occur phenomenon. In a
joint work with Kang Li and Huanxin Lin, we showed that every simple finite
Z-stable (not necessary unital) C*-algebra has stable rank one. And many
non-Z-stable C*-algebra (for example, Villadsen’s algebra of first type) also
has stable rank one. Recently, in a joint work with Huaxin Lin, we showed that
for a separable simple C*-algebra A which has comparison, A has stable rank one
is equivalent to has a tracial matricial structure, also equivalent to has
tracial approximately oscillation zero, and also equivalent to A has almost
stable rank one with the map $\Gamma: Cu(A)\to LAff_+(\tilde{QT}(A))$ is
surjective.
In this talk, I will talk about some recent results and some ideas of the proofs.