Cuntz Pimsner Algebra Research Articles

Splittings for $C^*$-correspondences and strong shift equivalence

We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to $C^*$-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of $C^*$-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular $C^*$-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant $*$-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for $C^*$-correspondences.

𝐶*-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces

In this paper we study Cuntz–Pimsner algebras associated to 𝐶*-correspondences over commutative C ∗ \mathrm {C}^* -algebras from the point of view of the C ∗ \mathrm {C}^* -algebra classification programme. We show that when the correspondence comes from an aperiodic homeomorphism of a finite dimensional infinite compact metric space X X twisted by a vector bundle, the resulting Cuntz–Pimsner algebras have finite nuclear dimension. When the homeomorphism is minimal, this entails classification of these C ∗ \mathrm {C}^* -algebras by the Elliott invariant. This establishes a dichotomy: when the vector bundle has rank one, the Cuntz–Pimsner algebra has stable rank one. Otherwise, it is purely infinite. For a Cuntz–Pimsner algebra of a minimal homeomorphism of an infinite compact metric space X X twisted by a line bundle over X X , we introduce orbit-breaking subalgebras. With no assumptions on the dimension of X X , we show that they are centrally large subalgebras and hence simple and stably finite. When the dimension of X X is finite, they are furthermore Z \mathcal {Z} -stable and hence classified by the Elliott invariant.

Subproduct systems with quantum group symmetry

We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones–Wenzl projections in Temperley–Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C ^* -algebras that are KK -equivalent to \mathbb C and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz–Pimsner algebras under monoidal equivalence of quantum symmetry groups.

Group Actions on Product Systems

We introduce the concept of a crossed product of a product system by a locally compact group. We prove that the crossed product of a row-finite and faithful product system by an amenable group is also a row-finite and faithful product system. We generalize a theorem of Hao and Ng about the crossed product of the Cuntz-Pimsner algebra of a $C^{\ast}$-correspondence by a group action to the context of product systems. We present examples related to group actions on $k$-graphs and to higher rank Doplicher-Roberts algebras.

Leavitt path algebras of labelled graphs

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.

Weighted Cuntz–Krieger Algebras

Let E be a finite directed graph with no sources or sinks and write \(X_E\) for the graph correspondence. We study the \(C^*\)-algebra \(C^*(E,Z):=\mathcal {T}(X_E,Z)/\mathcal {K}\) where \(\mathcal {T}(X_E,Z)\) is the \(C^*\)-algebra generated by weighted shifts on the Fock correspondence \(\mathcal {F}(X_E)\) given by a weight sequence \(\{Z_k\}\) of operators \(Z_k\in \mathcal {L}(X_{E^{k}})\) and \(\mathcal {K}\) is the algebra of compact operators on the Fock correspondence. If \(Z_k=I\) for every k, \(C^*(E,Z)\) is the Cuntz–Krieger algebra associated with the graph E. We show that \(C^*(E,Z)\) can be realized as a Cuntz–Pimsner algebra and use a result of Schweizer to find conditions for the algebra \(C^*(E,Z)\) to be simple. We also analyse the gauge-invariant ideals of \(C^*(E,Z)\) using a result of Katsura and conditions that generalize the conditions of subsets of \(E^0\) (the vertices of E) to be hereditary or saturated. As an example, we discuss in some details the case where E is a cycle.

The Koopman representation for self-similar groupoid actions

We introduce the [Formula: see text]-algebra [Formula: see text] generated by the Koopman representation [Formula: see text] of an étale groupoid [Formula: see text] acting on a measure space [Formula: see text]. We prove that for a level transitive self-similar action [Formula: see text] with [Formula: see text] finite and [Formula: see text] constant, there is an invariant measure [Formula: see text] on [Formula: see text] and that [Formula: see text] is residually finite dimensional with a normalized self-similar trace. We also discuss [Formula: see text]-fold similarities of Hilbert spaces in connection to representations of the graph algebra [Formula: see text] and self-similar representations of [Formula: see text] in connection to the Cuntz–Pimsner algebra [Formula: see text].

Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Motivated by the study of symmetries of C*-algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2)-equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz-Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K-theory. In particular, starting from an irreducible representation of SU(2), we show that the corresponding Toeplitz algebra is equivariantly KK-equivalent to the algebra of complex numbers. In this way, we obtain a six term exact sequence of K-groups containing a noncommutative analogue of the Euler class.

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

We establish exact sequences in KK-theory for graded relative Cuntz–Pimsner algebras associated to nondegenerate C⁎-correspondences. We use this to calculate the graded K-theory and K-homology of relative Cuntz–Krieger algebras of directed graphs for gradings induced by {0,1}–valued labellings of their edge sets.

Symmetries of the C⁎-algebra of a vector bundle

We consider C⁎-algebras constructed from compact group actions on complex vector bundles E→X endowed with a Hermitian metric. An action of G by isometries on E→X induces an action on the C⁎-correspondence Γ(E) over C(X) consisting of continuous sections, and on the associated Cuntz–Pimsner algebra OE, so we can study the crossed product OE⋊G. If the action is free and rank E=n, then we prove that OE⋊G is Morita–Rieffel equivalent to a field of Cuntz algebras On over the orbit space X/G. If the action is fiberwise, then OE⋊G becomes a continuous field of crossed products On⋊G. For transitive actions, we show that OE⋊G is Morita–Rieffel equivalent to a graph C⁎-algebra.

A von Neumann algebraic approach to self-similar group actions

We study some relations between self-similar group actions and operator algebras. We see that [Formula: see text] or [Formula: see text], where [Formula: see text] denotes the Bernoulli measure and [Formula: see text] the set of [Formula: see text]-generic points. In the case [Formula: see text], we get a unique KMS state for the canonical gauge action on the Cuntz–Pimsner algebra constructed from a self-similar group action by Nekrashevych. Moreover, if [Formula: see text], there exists a unique tracial state on the gauge invariant subalgebra of the Cuntz–Pimsner algebra. We also consider the GNS representation of the unique KMS state and compute the type of the associated von Neumann algebra.

The Cuntz-Pimsner extension and mapping cone exact sequences

For Cuntz-Pimsner algebras of bi-Hilbertian bimodules with finite Jones-Watatani index satisfying some side conditions, we give an explicit isomorphism between the $K$-theory exact sequences of the mapping cone of the inclusion of the coefficient algebra into a Cuntz-Pimsner algebra, and the Cuntz-Pimsner exact sequence. In the process we extend some results by the second author and collaborators from finite projective bimodules to certain finite index bimodules, and also clarify some aspects of Pimsner's `extension of scalars' construction.

A bicategorical interpretation for relative Cuntz-Pimsner algebras

We interpret the construction of relative Cuntz-Pimsner algebras of correspondences in terms of the correspondence bicategory, as a reflector into a certain sub-bicategory. This generalises a previous characterisation of absolute Cuntz-Pimsner algebras of proper correspondences as colimits in the correspondence bicategory.

Self-similark-Graph C*-Algebras

AbstractIn this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda $ and associate it a universal C$^\ast $-algebra ${{\mathcal{O}}}_{G,\Lambda }$. We prove that ${{\mathcal{O}}}_{G,\Lambda }$ can be realized as the Cuntz–Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${{\mathcal{O}}}_{G,\Lambda }$ is shown to be isomorphic to a “path-like” groupoid C$^\ast $-algebra. This facilitates studying the properties of ${{\mathcal{O}}}_{G,\Lambda }$. We show that ${{\mathcal{O}}}_{G,\Lambda }$ is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of ${{\mathcal{O}}}_{G,\Lambda }$ in terms of the underlying action, and we prove that, whenever ${{\mathcal{O}}}_{G,\Lambda }$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda $ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

Boundaries, spectral triples and $K$-homology

This paper extends the notion of a spectral triple to a relative spectral triple, an unbounded analogue of a relative Fredholm module for an ideal $J\triangleleft A$. Examples include manifolds with boundary, manifolds with conical singularities, dimension drop algebras, $\theta$-deformations and Cuntz–Pimsner algebras of vector bundles. The bounded transform of a relative spectral triple is a relative Fredholm module, making the image of a relative spectral triple under the boundary mapping in $K$-homology easy to compute. We introduce an additional operator called a Clifford normal with which a relative spectral triple can be doubled into a spectral triple. The Clifford normal also provides a boundary Hilbert space, a representation of the quotient algebra, a boundary Dirac operator and an analogue of the Calderon projection. In the examples this data does assemble to give a boundary spectral triple, though we can not prove this in general. When we do obtain a boundary spectral triple, we provide sufficient conditions for the boundary triple to represent the $K$-homological boundary. Thus we abstract the proof of Baum–Douglas–Taylor's "boundary of Dirac is Dirac on the boundary" theorem into the realm of non-commutative geometry.

Constructing KMS states from infinite-dimensional spectral triples

We construct KMS-states from Li1-summable semifinite spectral triples and show that in several important examples the construction coincides with well-known direct constructions of KMS-states for naturally defined flows. Under further summability assumptions the constructed KMS-state can be computed in terms of Dixmier traces. For closed manifolds, we recover the ordinary Lebesgue integral. For Cuntz–Pimsner algebras with their gauge flow, the construction produces KMS-states from traces on the coefficient algebra and recovers the Laca–Neshveyev correspondence. For a discrete group acting on its Stone–Čech boundary, we recover the Patterson–Sullivan measures on the Stone-Čech boundary for a flow defined from the Radon–Nikodym cocycle.

Poincaré duality for Cuntz–Pimsner algebras

We present a new approach to Poincaré duality for Cuntz–Pimsner algebras. We provide sufficient conditions under which Poincaré self-duality for the coefficient algebra of a Hilbert bimodule lifts to Poincaré self-duality for the associated Cuntz–Pimsner algebra.With these conditions in hand, we can constructively produce fundamental classes in K-theory for a wide range of examples. We can also produce K-homology fundamental classes for the important examples of Cuntz–Krieger algebras (following Kaminker–Putnam) and crossed products of manifolds by isometries, and their non-commutative analogues.

C∗-Algebras for partial product systems over N

We define partial product systems over N. They generalise product systems over N and Fell bundles over Z. We define Toeplitz C∗-algebras and relative Cuntz-Pimsner algebras for them and show that the section C∗-algebra of a Fell bundle over Z is a relative Cuntz-Pimsner algebra. We describe the gauge-invariant ideals in the Toeplitz C∗-algebra.

Topological freeness for C⁎-correspondences

We study conditions that ensure uniqueness theorems of Cuntz-Krieger type for relative Cuntz-Pimsner algebras $\mathcal{O}(J,X)$ associated to a $C^*$-correspondence $X$ over a $C^*$-algebra $A$. We give general sufficient conditions phrased in terms of a multivalued map $\widehat{X}$ acting on the spectrum $\widehat{A}$ of $A$. When $X(J)$ is of Type I we construct a directed graph dual to $X$ and prove a uniqueness theorem using this graph. When $X(J)$ is liminal, we show that topological freeness of this graph is equivalent to the uniqueness property for $\mathcal{O}(J,X)$, as well as to an algebraic condition, which we call $J$-acyclicity of $X$. As an application we improve the Fowler-Raeburn uniqueness theorem for the Toeplitz algebra $\mathcal{T}_X$. We give new simplicity criteria for $\mathcal{O}_X$. We generalize and enhance uniqueness results for relative quiver $C^*$-algebras of Muhly and Tomforde. We also discuss applications to crossed products by endomorphisms.

Cuntz-Pimsner algebras of group representations

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 ${\mathcal E}(\rho )={\mathcal H}\otimes _{\mathbb C} C^*(G)$ over $C^*(G)$ and study the Cuntz-Pimsner algebra ${\mathcal O}_{{\mathcal E}(\rho )}$. We prove that, for $G$ compact, ${\mathcal O}_{{\mathcal E}(\rho )}$ is strongly Morita equivalent to a graph $C^*$-algebra. If $\lambda $ is the left regular representation of an infinite, discrete and amenable group $G$, we show that ${\mathcal O}_{{\mathcal E}(\lambda )}$ is simple and purely infinite, with the same $K$-theory as $C^*(G)$. If $G$ is compact abelian, any representation decomposes into characters and determines a skew product graph. We illustrate with several examples, and we compare ${\mathcal E}(\rho )$ with the crossed product $C^*$-correspondence.