Real rank zero for purely infinite corona algebras

Ping Wong Ng

*(University of Louisiana at Lafayette)*

10:00 am - 11:00 am, May 10h, 2016 Science Building A1510

__Abstract:__

Let $A$ be a nonunital simple separable $C^*$-algebra with strict comparison, almost divisibility, stable rank one, and quasicontinuous scale.Then the corona algebra $M(A)/A$ has real rank zero.

The above generalizes a number of results in the literature.

The proof techniques are based on the following (earlier) result:

Theorem: Let $A$ be a nonunital simple separable $C^*$-algebra with strict comparison, almost divisibility,and stable rank one.Then the following statements are equivalent:

(i.) $A$ has quasicontinuous scale.

(ii.) $M(A)$ has strict comparison.

(iii.) $M(A)/A$ is purely infinite.

(iv.) $M(A)/I_{min}$ is purely infinite.

(v.) $M(A)$ has finitely many ideals.

(vi.) $I_{min} = I_{fin}$.

The above generalizes a number of results in the literature.

The proof techniques are based on the following (earlier) result:

Theorem: Let $A$ be a nonunital simple separable $C^*$-algebra with strict comparison, almost divisibility,and stable rank one.Then the following statements are equivalent:

(i.) $A$ has quasicontinuous scale.

(ii.) $M(A)$ has strict comparison.

(iii.) $M(A)/A$ is purely infinite.

(iv.) $M(A)/I_{min}$ is purely infinite.

(v.) $M(A)$ has finitely many ideals.

(vi.) $I_{min} = I_{fin}$.

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