# 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…

## 6 Citations

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

- MathematicsJournal of Mathematical Analysis and Applications
- 2021

### Homotopy of product systems and K-theory of Cuntz-Nica-Pimsner algebras

- MathematicsIndiana University Mathematics Journal
- 2022

We introduce the notion of a homotopy of product systems, and show that the Cuntz-Nica-Pimsner algebras of homotopic product systems over N^k have isomorphic K-theory. As an application, we give a…

### The Cayley transform in complex, real and graded K-theory

- Mathematics
- 2019

We use the Cayley transform to provide an explicit isomorphism at the level of cycles from van Daele $K$-theory to $KK$-theory for graded $C^*$-algebras with a real structure. Isomorphisms between…

### Honours thesis: Exact sequences in graded $KK$-theory for Cuntz-Pimsner algebras

- Mathematics
- 2020

In this thesis we generalise the six-term exact sequence in graded $KK$-theory obtained in a paper of Kumjian, Pask and Sims (2017) to allow correspondences with non-compact left action. In…

### K-theory and homotopies of twists on ample groupoids

- MathematicsJournal of Noncommutative Geometry
- 2021

This paper investigates the K-theory of twisted groupoid C*-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum–Connes conjecture with coefficients gives rise to…

### $\mathrm{K}$-theory and homotopies of twists on ample groupoids

- Mathematics
- 2019

This paper investigates the $\mathrm{K}$-theory of twisted groupoid $\mathrm{C}^*$-algebras. It is shown that a homotopy of twists on an ample groupoid satisfying the Baum-Connes conjecture with…

## References

SHOWING 1-10 OF 44 REFERENCES

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

- Mathematics
- 1999

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

- Mathematics
- 2010

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

- MathematicsDocumenta Mathematica
- 2014

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…

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

- Mathematics
- 1981

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

- Mathematics
- 2011

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

- Mathematics
- 2013

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

- Mathematics
- 2007

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…

### The Toeplitz algebra of a Hilbert bimodule

- Mathematics
- 1998

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz…

### On higher rank graph C ∗ -algebras

- Mathematics
- 2000

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C…