Graphs of $C^*$-correspondences and Fell bundles

@article{Deaconu2008GraphsO,
  title={Graphs of \$C^*\$-correspondences and Fell bundles},
  author={Valentin Deaconu and Alex Kumjian and David Pask and Aidan Sims},
  journal={arXiv: Operator Algebras},
  year={2008}
}
We define the notion of a $\Lambda$-system of $C^*$-correspondences associated to a higher-rank graph $\Lambda$. Roughly speaking, such a system assigns to each vertex of $\Lambda$ a $C^*$-algebra, and to each path in $\Lambda$ a $C^*$-correspondence in a way which carries compositions of paths to balanced tensor products of $C^*$-correspondences. Under some simplifying assumptions, we use Fowler's technology of Cuntz-Pimsner algebras for product systems of $C^*$-correspondences to associate a… 
View PDF on arXiv
17 Citations
Highly Influential Citations
3
Background Citations
7
Methods Citations
3

17 Citations

Groupoid actions on $C^*$-correspondences

Let the groupoid $G$ with unit space $G^0$ act via a representation $\rho$ on a $C^*$-correspondence ${\mathcal H}$ over the $C_0(G^0)$-algebra $A$. By the universal property, $G$ acts on the…

Group actions on graphs and $C^*$-correspondences

If $G$ acts on a $C^*$-correspondence ${\mathcal H}$, then by the universal property $G$ acts on the Cuntz-Pimsner algebra ${\mathcal O}_{\mathcal H}$ and we study the crossed product ${\mathcal…

Cuntz-Pimsner algebras of group representations

  • V. Deaconu
  • Mathematics
    Rocky Mountain Journal of Mathematics
  • 2018
Given a locally compact group $G$ and a unitary representation $\rho:G\to U({\mathcal H})$ on a Hilbert space ${\mathcal H}$, we construct a $C^*$-correspondence ${\mathcal E}(\rho)={\mathcal…

Semicrossed Products of Operator Algebras by Semigroups

We examine the semicrossed products of a semigroup action by $*$-endomorphisms on a C*-algebra, or more generally of an action on an arbitrary operator algebra by completely contractive…

On Nica-Pimsner algebras of C*-dynamical systems over $\mathbb{Z}_+^n$

We examine Nica-Pimsner algebras associated with semigroup actions of $\mathbb{Z}_+^n$ on a C*-algebra $A$ by $*$-endomorphisms. We give necessary and sufficient conditions on the dynamics for…

Contributions to the theory of C*-correspondences with applications to multivariable dynamics

Motivated by the theory of tensor algebras and multivariable C*-dynamics, we revisit two fundamental techniques in the theory of C*-correspondences, the "addition of a tail" to a non-injective…

C⁎-algebras and equivalences for C⁎-correspondences

Product systems and their representations: an approach using Fock spaces and Fell bundles

In this exposition we highlight product systems as the semigroup analogue of Fell bundles. Motivated by Fock creation operators we extend the definition of Fowler’s product systems over unital…

Operator algebras for higher rank analysis and their application to factorial languages

We study strong compactly aligned product systems of ℤ+N over a C*-algebra A. We provide a description of their Cuntz-Nica-Pimsner algebra in terms of tractable relations coming from ideals of A.…

Operator algebras for higher rank analysis and their application to factorial languages

We study strong compactly aligned product systems of ℤ + N over a C*-algebra A . We provide a description of their Cuntz-Nica-Pimsner algebra in terms of tractable relations coming from ideals of A .…

References

SHOWING 1-10 OF 70 REFERENCES

$C^{\ast}$-Algebras associated with Mauldin-Williams Graphs

A Mauldin-Williams graph $\mathcal{M}$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a…

Asymptotic K-Theory for Groups Acting on à 2 Buildings

Abstract Let $\Gamma$ be a torsion free lattice in $G=\text{PGL}\left( 3,\mathbb{F} \right)$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits…

C*-Algebras Associated with Mauldin–Williams Graphs

Abstract A Mauldin–Williams graph $M$ is a generalization of an iterated function system by a directed graph. Its invariant set $K$ plays the role of the self-similar set. We associate a ${{C}^{*}}$…

Actions of $\mathbb{Z}^k$ associated to higher rank graphs

An action of $\mathbb{Z}^k$ is associated to a higher rank graph $\Lambda$ satisfying a mild assumption. This generalizes the construction of a topological Markov shift arising from a non-negative…

Multiply generated dynamical systems and the duality of higher rank graph algebras

We define a semidirect product groupoid of a system of partially defined local homeomorphisms $T=(T_{1},..., T_{r})$. We prove that this construction gives rise to amenable groupoids. The associated…

Fell bundles over groupoids

We study the C*-algebras associated to Fell bundles over groupoids and give a notion of equivalence for Fell bundles which guarantees that the associated C*-algebras are strong Morita equivalent. As…

Entropy of shifts on higher-rank graph C*-algebras

Let O_{Lambda} be a higher rank graph C*-algebra of rank r. For every tuple p of non-negative integers there is a canonical completely positive map Phi^p on O_{Lambda} and a subshift T^p on the path…

HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS

Abstract We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. We describe a variant of the Cuntz–Krieger relations which applies to…

Cuntz-Pimsner $C*$-Algebras and Crossed Products by Hilbert $C*$-Bimodules

Given a correspondence X over a C*-algebra A, we construct a C*-algebra and a Hilbert C*-bimodule over it whose crossed product is isomorphic to the augmented Cuntz-Pimsner C*-algebra of X. This…

Exel's crossed product and relative Cuntz–Pimsner algebras

We consider Exel's new construction of a crossed product of a $C^*$-algebra $A$ by an endomorphism $\alpha$. We prove that this crossed product is universal for an appropriate family of covariant…
...