Tracial approximate divisibility and stable rank one

Xuanlong Fu ·ûÐþÁú

*(University of Toronto)*

10:00-11:00, November 22, 2021 Tencent Meeting£º688 6335 2402

__Abstract:__

In 2004, M. Rordam showed that every unital simple finite ${\cal Z}$-stable C*-algebra has stable rank one. The question that whether a non-unital simple finite ${\cal Z}$-stable C*-algebra has stable rank one or not is remained open since then.

Recently we show that every separable simple tracially approximately divisible C*-algebra has strict comparison, and, it is either purely infinite or has stable rank one.

As a consequence, we show that every (non-unital) simple finite ${\cal Z}$-stable C*-algebra has stable rank one. Hence the problem of dichotomy of finiteness and of stable rank for simple ${\cal Z}$-stable (not necessary unital) C*-algebras is settled. This is a joint work with Kang Li and Huaxin Lin.

Recently we show that every separable simple tracially approximately divisible C*-algebra has strict comparison, and, it is either purely infinite or has stable rank one.

As a consequence, we show that every (non-unital) simple finite ${\cal Z}$-stable C*-algebra has stable rank one. Hence the problem of dichotomy of finiteness and of stable rank for simple ${\cal Z}$-stable (not necessary unital) C*-algebras is settled. This is a joint work with Kang Li and Huaxin Lin.

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