# Simple stably projectionless C*-algebras with generalized tracial rank one

@article{Elliott2017SimpleSP,
title={Simple stably projectionless C*-algebras with generalized tracial rank one},
author={George A. Elliott and Guihua Gong and Huaxin Lin and Zhuang Niu},
journal={arXiv: Operator Algebras},
year={2017}
}
We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. Some structural question concerning these simple C*-algebras are studied. The paper also serves as a technical support for the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras.
16 Citations
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#### References

SHOWING 1-10 OF 60 REFERENCES
On the positive tracial cones of simple stably projectionless C*-algebras
Abstract It is shown that any topological cone, with a base consisting of a metrizable Choquet simplex, arises as the positive tracial cone of a simple stably projectionless C * -algebra inductiveExpand
On the Classification of Simple Stably Projectionless C*-Algebras
Abstract It is shown that simple stably projectionless ${{\text{C}}^{*}}$ -algebras which are inductive limits of certain specified building blocks with trivial $\text{K}$ -theory are classified byExpand
A simple, monotracial, stably projectionless C*-algebra
It is shown that every nondegenerate endomorphism of W is approximately inner and a trace-preserving embedding of W into the central sequences algebra M(W)_\infty \cap W'. Expand
Simple corona C^*-algebras
Let A be a non-unital and σ-unital simple C*-algebra. We show that if M(A)/A is simple, then M(A)/A is purely infinite. We also show that M(A)/A is simple if and only if A has a continuous scaleExpand
Quasidiagonality of nuclear C*-algebras
• Mathematics
• 2015
We prove that faithful traces on separable and nuclear C*-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification ofExpand
Simple *-algebras with continuous scales and simple corona algebras
It is shown that the corona algebra M(A)/A of a separable simple C -algebra A is simple if and only if A has a continuous scale or A is elementary. It is also shown that simple C -algebras withExpand
On the classification of simple amenable C*-algebras with finite decomposition rank
• Mathematics
• 2015
Let $A$ be a unital simple separable C*-algebra satisfying the UCT. Assume that $\mathrm{dr}(A)<+\infty$, $A$ is Jiang-Su stable, and $\mathrm{K}_0(A)\otimes \mathbb{Q}\cong \mathbb{Q}$. Then $A$ isExpand
Cuntz semigroups of C*-algebras of stable rank one and projective Hilbert modules
Let $A$ be a simple C*-algebra of stable rank one and let $p$ and $q$ be two $\sigma$-compact open projections. It is proved that there is a continuous path of unitaries in ${\tilde A}$ whichExpand
The Jiang–Su algebra revisited
• Mathematics
• 2008
Abstract We give a number of new characterizations of the Jiang–Su algebra 𝒵, both intrinsic and extrinsic, in terms of C*-algebraic, dynamical, topological and K-theoretic conditions. Along the wayExpand
The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension
• Mathematics
• 2017
The class of simple separable KK-contractible (KK-equivalent to $\{0\}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the classExpand