Graded C*-algebras, Graded K-theory, And Twisted P-graph C*-algebras

@article{Kumjian2017GradedCG,
  title={Graded C*-algebras, Graded K-theory, And Twisted P-graph C*-algebras},
  author={Alex Kumjian and David Pask and Aidan Sims},
  journal={arXiv: Operator Algebras},
  year={2017},
  pages={295}
}
We develop methods for computing graded K-theory of C*-algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner-Voiculescu sequence. We introduce the notion of a twisted P-graph C*-algebra and establish connections with graded C*-algebras. Specifically, we show how a functor from a P-graph into the group of order two determines a grading of the… 
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6 Citations
Background Citations
3
Results Citations
2

6 Citations

Graded K-theory and K-homology of relative Cuntz–Pimsner algebras and graph C⁎-algebras

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References

SHOWING 1-10 OF 44 REFERENCES

On twisted higher-rank graph C*-algebras

We define the categorical cohomology of a k-graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This…

Homology for higher-rank graphs and twisted C*-algebras

ON Z/2Z-GRADED KK-THEORY AND ITS RELATION WITH THE GRADED Ext-FUNCTOR

This paper studies the relation between KK-theory and the Ext- functor of Kasparov for Z2-graded C -algebras. We use an approach similar to the picture of J. Cuntz in the ungraded case. We show that…

Co-universal C*-algebras associated to generalised graphs

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in ℕ. We focus on semigroups P arising as part of a…

Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish…

Exact Sequences for the Kasparov Groups of Graded Algebras

In [11] G. G. Kasparov defined the “operator K-functor” KK(A, B) associated with the graded C*-algebras A and B. If the algebras A and B are trivially graded and A is nuclear he proves six term exact…

THE OPERATOR K-FUNCTOR AND EXTENSIONS OF C*-ALGEBRAS

In this paper a general operator K-functor is constructed, depending on a pair A, B of C*-algebras. Special cases of this functor are the ordinary cohomological K-functor K*(B) and the homological…

HOMOLOGY FOR HIGHER-RANK GRAPHS AND TWISTED

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homol- ogy of a k-graph coincides with the…

Twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs

To each finitely aligned higher-rank graph $\Lambda$ and each $\mathbb{T}$-valued 2-cocycle on $\Lambda$, we associate a family of twisted relative Cuntz-Krieger algebras. We show that each of these…

Crossed products of k-graph C*-algebras by Zl

An action of Zl by automorphisms of a k-graph induces an action of Zl by automorphisms of the corresponding k-graph C*-algebra. We show how to construct a (k + l)-graph whose C*-algebra coincides…